GIFT  OF 
the  estate  of 

Professor  William  F,  Keyer 


rx-^^o/  / 


^.^- 


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in  2007  with  funding  from 

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THE   MODERN  MATHEMATICAL  SERIES 

LUCIEN  AUGUSTUS  WAIT  .  .  .  General  Editor 

(SBNIOB  PEOFE880K  OF  MATHEMATICS  IN  OOENELL  UNIVEKSITT) 


The  Modern  Mathematical  Series, 
lucien  augustus  wait, 

(Senior  Professor  of  Matbematics  in  Cornell  University,) 
GENERAL  EDITOR. 


This  series  includes  the  following  works : 

ANALYTIC  GEOMETRY.    By  J.  H.  Tanner  and  Joseph  Allen. 

DIFFERENTIAL  CALCULUS.    By  James  McMahon  and  Virgil  Snyj)er. 

INTEGRAL  CALCULUS.    By  D.  A.  Murray. 

DIFFERENTIAL  AND  INTEGRAL  CALCULUS.    By  Virgil  Snyder  and  J.  I 
Hutchinson. 

ELEMENTARY  ALGEBRA.    By  J.  H.  Tanner. 

ELEMENTARY  GEOMETRY.    By  James  McMahon. 


The  Analytic  Geometry,  Dififerential  Calculus,  and  Integral  Calculus  (pub- 
lished in  September  of  1898)  were  written  primarily  to  meet  the  needs  of  college 
students  pursuing  courses  in  Engineering  and  Architecture ;  accordingly,  prac- 
tical problems,  in  illustration  of  general  principles  under  discussion,  play  an 
important  part  in  each  book. 

These  three  books,  treating  their  subjects  in  a  way  that  is  simple  and  practi- 
cal, yet  thoroughly  rigorous,  and  attractive  to  both  teacher  and  student,  received 
such  general  and  hearty  approval  of  teachers,  and  have  been  so  widely  adopted 
in  the  best  colleges  and  universities  of  the  country,  that  other  books,  written  on 
the  same  general  plan,  are  being  added  to  the  series. 

The  Differential  and  Integral  Calculus  in  one  volume  was  written  especially 
for  those  institutions  where  the  time  given  to  these  subjects  is  not  sufficient  to 
use  advantageously  the  two  separate  books. 

The  more  elementary  books  of  this  series  are  designed  to  implant  the  spirit  of 
the  other  books  into  the  secondary  schools.  This  will  make  the  work,  from  the 
schools  up  through  the  university,  continuous  and  harmonious,  and  free  from 
the  abrupt  transition  which  the  student  so  often  experiences  in  changing  from 
his  preparatory  to  his  college  mathematics. 


DIFFERENTIAL  AND  INTEGRAL 
CALCULUS 


BY 

VIRGIL   SNYDER,   Ph.D.  (gottingen) 

AN© 

JOHN  IRWIN  HUTCHINSON,  Ph.D.  (chicago) 

OF  CORNELL  UNIVERSITY 


3jO<C 


NEW  YORK  .:•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


COPTEIGHT,   1902,    BY 

VIRGIL  SNYDER  and  JOHN  I.   HUTCHINSON 

EXTSEED  AT  StATIONKRS'    HaLL,   LONDON. 


DiF.  nrr.  oal. 
w.  p.  5 


QA303 

SGI 


PREFACE 

The  favorable  reception  accorded  the  two  volumes  on  the 
Calculus  in  this  series  shows  that  they  have  been  serviceable 
in  supplying  a  real  need.  A  general  demand  has  arisen  for 
a  similar  treatment  of  the  subjects  in  briefer  form,  suitable 
for  use  in  shorter  and  more  elementary  courses.  Accord- 
ingly, in  response  to  numerous  requests  and  suggestions,  the 
present  volume  has  been  prepared. 

The  part  on  the  Differential  Calculus  is  of  essentially  the 
same  character  as  the  former  separate  volume  (which  will 
be  referred  to  in  the  text  as  D.  C),  but  the  range  of  topics 
is  restricted ;  various  theorems  have  been  put  in  less  ab- 
stract form,  and  fewer  alternative  proofs  have  been  given. 
The  chapter  on  the  expansion  of  functions  has  been  so 
arranged  that  the  remainder  theorem  may  be  omitted  with- 
out marring  the  continuity  of  the  subject.  In  the  treatment 
of  functions  of  two  independent  variables  no  use  is  made  of 
an  auxiliary  variable. 

The  characteristic  features  of  the  larger  book  are  retained. 
Some  of  these  are  as  follows :  — 

1.  The  derivative  is  presented  rigorously  as  a  limit. 

2.  The  process  of  differentiation  is  so  arranged  as  to  give 
the  a;-derivative  of  a  function  of  u^  in  which  t^  is  a  function 
of  X ;  the  resulting  type  forms  being  printed  in  full-face 
letters  in  the  text  and  collected  for  reference  at  the  end  of 
the  chapter. 


mstts^s 


VI  PREFACE 

3.  Maxima  and  minima  are  discussed  as  the  turning 
values  in  the  variation  of  a  function,  with  complete  graphi- 
cal representation. 

4.  The  notions  of  rates  and  differentials  are  so  presented 
as  to  grow  naturally  out  of  the  idea  of  a  derivative,  and  are 
not  introduced  until  the  student  has  become  familiar  with 
the  process  of  finding  the  derivative  and  with  its  use  in 
studying  the  variation  of  a  function. 

5.  The  related  theories  of  inflexions,  curvature,  and 
asymptotes  receive  direct  and  comprehensive  treatment. 

The  part  on  the  Integral  Calculus  has  been  written,  en- 
tirely anew. 

The  first  five  chapters  discuss  the  ordinary  methods  of 
integration.  The  aim  has  been  to  make  clear  the  rationale 
of  each  process,  and  to  encourage  the  students  to  become 
independent  of  formulas. 

The  method  of  reduction  has  been  put  in  the  simplest 
possible  form  ;  in  the  solution  of  problems  students  need 
make  no  use  of  formulas  of  reduction. 

In  the  resolution  of  rational  fractions  into  simpler  ones, 
care  has  been  taken  to  show  the  logical  basis  of  the  usual 
assumptions. 

The  rationalization  of  a  differential  containing  the  square 
root  of  a  quadratic  expression  has  been  treated  much  more 
fully  than  usual.  The  problem  is  interpreted  geometrically 
as  equivalent  to  the  rational  expression  of  the  coordinates  of 
a  variable  point  on  a  conic  in  terms  of  a  varying  parameter. 
This  makes  clear  how  the  required  transformations  are  sug- 
gested and  puts  the  subject  in  a  more  attractive  form. 

Special  care  has  been  taken  in  presenting  the  subject  of 
integration  regarded  as  a  summation  so  as  to  combine  rigor 
and  simplicity.     The  ordinary  cases  of  discontinuity,  either 


PREFACE  Vll 

of  the  integrand  or  of  the  variable  of  integration,  are  in- 
cluded in  the  discussion. 

In  deriving  the  formula  for  length  of  arc,  the  definition 
of  such  length  is  given  as  the  limit  of  the  sum  of  chords,  a 
definition  which  readily  expresses  itself,  by  the  use  of  the 
mean- value  theorem,  in  the  form  of  a  definite  integral. 

The  exercises,  which  are  new  throughout  the  book,  are 
carefully  graded.  Numerous  illustrative  examples  are  worked 
out  in  the  text,  and  are  accompanied  by  various  suggestions 
and  remarks  relating  to  both  theory  and  practice. 

The  authors  gratefully  acknowledge  their  indebtedness  to 
their  colleague,  Professor  James  McMahon,  for  permission 
to  make  free  use  of  McMahon  and  Snyder's  Differential 
Calculus,  for  a  number  of  valuable  suggestions,  and  for  as- 
sistance in  reading  portions  of  the  manuscript  and  proof. 


CONTENTS 

DIFFERENTIAL   CALCULUS 

CHAPTER  I 
Fundamental  Principles 

ARTICLE  PAGE 

1.  Elementary  definitions 1 

2.  Infinitesimals  and  infinites 2 

3.  Fundamental  theorems  concerning  infinitesimals  and  limits  in 

general 4 

4.  Comparison  of  variables 5 

5.  Comparison  of  infinitesimals  and  of  infinites.     Orders  of  mag- 

nitude         7 

6.  Useful  illustrations  of  infinitesimals  of  different  orders     .         .  9 

7.  Continuity  of  functions 13 

8.  Comparison  of  simultaneous  infinitesimal  increments  of  two 

related  variables 15 

9.  Definition  of  a  derivative 20 

10.  Geometrical  illustrations  of  a  derivative 20 

11.  The  operation  of  differentiation 24 

12.  Increasing  and  decreasing  functions 25 

13.  Algebraic  test  of  the  intervals  of  increasing  and  decreasing      .  27 

14.  Differentiation  of  a  function  of  a  function          ....  28 

CHAPTER  II 

Differentiation  of  the  Elementary  Forms 

15.  Differentiation  of  the  product  of  a  constant  and  a  variable       .  30 

16.  Differentiation  of  a  sum 31 

17.  Differentiation  of  a  product 32 

18.  Differentiation  of  a  quotient 33 

19.  Differentiation  of  a  commensurable  power  of  a  function   .         .  34 

20.  Elementary  transcendental  functions 38 

21.  Differentiation  of  log^  x  and  log,,  u 39 

ix 


X  CONTENTS 

ARTICLE  PAGS 

22.  Differentiation  of  the  simple  exponential  function    ...  41 

23.  Differentiation  of  the  general  exponential  function    ...  42 

24.  Differentiation  of  an  incommensurable  power    ....  42 

25.  Differentiation  of  sin  u ^        .  44 

26.  Differentiation  of  cos  u .  44 

27.  Differentiation  of  tan  u        .    ' 45 

28.  Differentiation  of  cot  m 45 

29.  Differentiation  of  sec  m 46 

30.  V  Differentiation  of  esc  m 46 

31.  Differentiation  of  vers  u 46 

32.  Differentiation  of  sin-^  u 47 

33.  Differentiation  of  the  remaining  inverse  trigonometric  forms  .  48 

34.  Table  of  fundamental  forms .  49 

CHAPTER  III 
Successive  Differentiation 

35.  Definition  of  the  wth  derivative ,  52 

36.  Expression  for  the  nth  derivative  in  certain  cases      ...  54 

CHAPTER  IV 
Expansion  of  Functions 

37.  Convergence  and  divergence  of  series          .....  57 

38.  General  test  for  interval  of  convergence 58 

39.  Remainder  after  n  terms 61 

40.  Maclaurin's  expansion  of  a  function  in  power-series  ...  62 

41.  Taylor's  series 66 

42.  Rolle's  theorem 67 

43.  Form  of  remainder  in  Maclaurin's  series 68 

44.  Another  expression  for  remainder 70 

45.  Theorem  of  mean  value 75 

CHAPTER  V 
Indeterminate  Forms 

46.  Definition  of  an  indeterminate  form 77 

47.  Indeterminate  forms  may  have  determinate  values    ...  78 

48.  Evaluation  by  development 80 

49.  Evaluation  by  differentiation 81 


CONTENTS  XI 

ARTICLE  PAGE 

50.  Evaluation  of  the  indeterminate  form  §- 84 

51.  Evaluation  of  the  form  co  .  0 86 

52.  Evaluation  of  the  form  go  —  co.        .        •        ••        .        .86 

53.  Evaluation  of  the  form  1" 88 

54.  Evaluation  of  the  forms  0^,00^ 88 


CHAPTER  VI 
Mode  of  Variation  of  Functions  of  One  Variable 

55.  Review  of  increasing  and  decreasing  functions  ....  91 

56.  Turning  values  of  a  function 91 

57.  Critical  values  of  the  variable 93 

58.  Method  of   determining  whether  </>'  (x)  changes  its  sign  in 

passing  through  zero  or  infinity 93 

59.  Second  method  of  determining  whether  <^'  {x)  changes  sign  in 

passing  through  zero 95 

60.  Conditions  for  maxima  and  minima  derived  from   Taylor's 

theorem 97 

61.  The  maxima  and  minima  of  any  continuous  function  occur 

alternately 98 

62.  Simplifications  that  do  not  alter  critical  values  ...      99 

63.  Greometric  problems  in  maxima  and  minima     ....    100 

CHAPTER  Vn 
Rates  and  Differentials 

64.  Rates.     Time  as  independent  variable 105 

65.  Abbreviated  notation  for  rates 108 

66.  Differentials  often  substituted  for  rates 110 

CHAPTER  Vni 
Differentiation  of  Functions  of  Two  Variables 

67.  Definition  of  continuity 113 

68.  Partial  differentiation 114 

69.  Total  differential 116 

70.  Language  of  differentials 119 

71.  Differentiation  of  implicit  functions  ......  120 

72.  Successive  partial  differentiation         ......  121 

73.  Order  of  differentiation  indifferent     .        .        .        •        .        .121 


XU  CONTENTS 

CHAPTER  IX 
Change  of  the  Variable 

ARTIOLK  PAOB 

74.  Interchange  of  dependent  and  independent  variables         .        .    124 

75.  Change  of  the  dependent  variable 125 

76.  Change  of  the  independent  variable 126 

APPLICATIONS  TO  GEOMETRY 

CHAPTER  X 

Tangents  and  Normals 

77.  Geometric  meaning  oi-^ -       %   ^29 

78.  Equation  of  tangent  and  normal  at  a  given  point       .        .        .    129 

79.  Length  of  tangent,  normal,  subtangent,  subnormal    .        .        .    130 

Polar  Coordinates 

80.  Meaning  of  p^ 133 

81.  Relation  between  -~  and  p-r 134 

ax         ^  dp 

82.  Length  of  tangent,  normal,  polar  subtangent,  and  polar  sub- 

normal  135 

CHAPTER  XI 

Derivative  of  an  Arc,  Area,  Volume,  and  Surface 
OF  Revolution 

83.  Derivative  of  an  arc 138 

84.  Trigonometric  meaning  of  — ,   -r- 139 

85.  Derivative  of  the  volume  of  a  solid  of  revolution       .        .        .  140 

86.  Derivative  of  a  surface  of  revolution 140 

87.  Derivative  of  arc  in  polar  coordinates 141 

88.  Derivative  of  area  in  polar  coordinates 142 

CHAPTER  XH 
Asymptotes 

89.  Hyperbolic  and  parabolic  branches 143 

90.  DefinitiGD  of  a  rectilinear  asymptote 143 


C0:N  TENTS  XIU 
Determination  of  Asymptotes 

ARTICLE  PAGE 

91.  Method  of  limiting  intercepts 143 

92.  Method  of  inspection.     Infinite  oi'dinates,  asymptotes  parallel 

to  axes 144 

93.  Method  of  substitution.     Oblique  asymptotes  ....  147 

94.  Number  of  asymptotes 149 

95.  Method  of  expansion.     Explicit  functions        ....  150 

CHAPTER  Xin 
Direction  of  Bending.    Points  of  Inflexion 

96.  Concavity  upward  and  downward     .         .         .         .        .         .  152 

97.  Algebraic  test  for  positive  and  negative  bending      .         .         .  153 

98.  Analytical  derivation  of  the  test  for  the  direction  of  bending .  156 

99.  Concavity  and  convexity  towards  the  axis        ....  157 

CHAPTER  XIV 
Contact  and  Curvature 

100.  Order  of  contact .159 

101.  Number  of  conditions  implied  by  contact         ....  160 

102.  Contact  of  odd  and  of  even  order 161 

103.  Circle  of  curvature 163 

104.  Length   of   radius  of    curvature;    coordinates  of   center  of 

curvature 163 

105.  Direction  of  radius  of  curvature 164 

106.  Total  curvature  of  a  given  arc ;  average  curvature  .        .        .  166 

107.  Measure  of  curvature  at  a  given  point 166 

108.  Curvature  of  osculating  circle 167 

109.  Direct  derivation  of  the  expression  for  k  and  E  in  polar 

coordinates 169 

EVOLUTES  AND   INVOLUTES 

110.  Definition  of  an  evolute     . 170 

111.  Properties  of  the  evolute 172 

CHAPTER  XV 
Singular  Points 

112.  Definition  of  a  singular  point 179 

113.  Determination  of  singular  points  of  algebraic  curves       o        .  179 


XIV  CONTENTS 

ARTICLE  PAGB 

114.  Multiple  points 181 

115.  Cusps 182 

116.  Conjugate  points 184 

CHAPTER  XVI 
Envelopes 


117.  Family  of  curves 

118.  Envelope  of  a  family  of  curves  .... 

119.  The  envelope  touches  every  curve  of  the  family 

120.  Envelope  of  normals  of  a  given  curve 

121.  Two  parameters,  one  equation  of  condition 


187 
188 
189 
190 
191 


INTEGRAL   CALCULUS 

CHAPTER  I 
General  Principles  of  Integration 

122.  The  fundamental  problem 195 

123.  Integration  by  inspection 196 

124.  The  fundamental  formulas  of  integration         ....  198 

125.  Certain  general  principles 199 

126.  Integration  by  parts 203 

127.  Integration  by  substitution 205 

128.  Additional  standard  forms 209 

129.  Integrals  of  the  form  f  M^  +  -g)^£ 210 

*"  y/ax^  -{-bx+  c 

130.  Integrals  of  the  form  f- '^^  ...    212 

JiAx  +  B)  Vax2  -i-bx-^c 

CHAPTER  n 

131.  Reduction  Formulas 215 

CHAPTER  m 
Integration  op  Rational  Fractions 

132.  Decomposition  of  rational  fractions 223 

183.  Case  L    Factors  of  the  first  degree,  none  repeated         .        .    225 


CONTENTS  XV 

ARTICLE                                                                                                                                                 -  PAGE 

134.  Case  II.    Factors  of  the  first  degree,  some  repeated       .        .  226 

135.  Case  III.     Occurrence  of  quadratic  factors,  none  repeated     .  228 

136.  Case  IV.     Occurrence  of  quadratic  factors,  some  repeated     .  229 

137.  General  theorem  on  the  integration  of  rational  fractions         .  230 

CHAPTER  IV 
Integration  by  Rationalization 

138.  Integration  of  functions  containing  the  irrationality  Vax  -\-  b  231 

139.  Integration  of  expressions  containing  Vax^  -\-hx  +  c        .         .  232 

140.  General  theorem  on  the  integration  of  irrational  functions      .  236 

CHAPTER  V 

Integration  of  Trigonometric  and  Other  Tran- 
scendental Functions 

141.  Integration  by  substitution 238 

142.  Integration  of  i  sec^"  x  dx,     j  cosec^**  xdx          .        .        .        .  238 

143.  Integration  of  J  sec* x tan^" +'^xdx,     \  coseC" x cot^™ ^'^xdx      .  239 

144.  Integration  of  \  tan« x dx,    \  cot" xdx 240 

145.  Integration  of  j  sin^a:  cos"a:c?a: 242 


244 


246 


146.  Integration  of  ( ; ,     i t—. —  .        .        .        , 

J  a-\-o  cos  X     J  a  -{•  0  sm  x 

147.  Integration  of  J  e«=*  sin  nx  dx,     \  e"^  cos  nxdx      .   '     . 

CHAPTER  VI 
Integration  as  a  Summation 

148.  The  definite  integral 248 

149.  Geometrical  interpretation  of  the  definite  integral  as  an  area     253 

150.  Generalization  of  the  area  formula.     Positive  and  negative 

area 255 

151.  Certain  properties  of  definite  integrals 256 

152.  Definition  of  the  definite  integral  when  f(x)  becomes  infinite. 

Infinite  limits 257 


XVi  CONTENTS 


CHAPTER  VII 
Geometrical  Applications 

ARTICLE  PAGB 

153.  Areas.    Rectangular  coordinates 260 

154.  Areas.     Second  method 260 

155.  Precautions  to  be  observed  in  evaluating  definite  integrals      .  263 

156.  Areas.     Polar  coordinates 267 

157.  Length  of  curves.     Rectangular  coordinates     ....  269 

158.  Length  of  curves.     Polar  coordinates 271 

159.  Measurement  of  arcs  by  the  aid  of  parametric  representation  .  273 

160.  Area  of  surface  of  revolution     . 274 

161.  Volume  of  solid  of  revolution *  .        .277 

162.  Miscellaneous  applications .        .  279 

CHAPTER  VIII 
Successive  Integration 

163.  Successive  integration  of  functions  of  a  single  variable  .        .  286 

164.  Integration  of  functions  of  several  variables     ....  288 

165.  Integration  of  a  total  differential 289 

166.  Multiple  integrals .292 

167.  Definite  multiple  integrals 293 

168.  Plane  areas  by  double  integration 294 

169.  Volumes 295 


DIFFERENTIAL   CALCULUS 


3j»iC 


CHAPTER   I 

FUNDAMENTAL  PRINCIPLES 

1.  Elementary  definitions.  A  constant  number  is  one  that 
retains  the  same  value  throughout  an  investigati6n  in  which 
it  occurs.  A  variable  number  is  one  that  changes  from  one 
value  to  another  during  an  investigation.  When  the  varia- 
tion of  a  number  can  be  assigned  at  will,  the  variable  is  called 
independent;  when  the  value  of  one  number  is  determined 
by  that  of  another,  the  former  is  called  a  dependent  variable. 
The  dependent  variable  is  called  a  function  of  the  indepen- 
dent  variable. 


E.g.,  3  x%  4  Va;  —  1,  cos  x,  are  all  functions  of  x. 

Functions  of  one  variable  x  will  be  denoted  by  the  sym- 
bols /(a;),  <i>(x)y  •••;  similarly,  if  2  be  a  function  of  two 
variables  a?,  ^,  it  will  be  denoted  by  such  expressions  as 

z  =f(P^^  y')^  2J  =  F(x,  y)  '". 
When  a  variable  approaches  a  constant  in  such  a  way  that 
the  difference  between  the  variable  and  the  constant  may 
become  and  remain  smaller  than  any  fixed  number,  pre- 
viously assigned,  the  constant  is  called  the  limit  of  the 
variable. 

There  is  nothing  in  this  definition  which  requires  a  vari- 
able to  attain  the  value  of  its  limit,  or  not  to  attain  it.     The 

1 


2  DIFFERENTIAL   CALCULUS  [Ch.  I. 

examples  of  limits  met  with  in  elementary  geometry  are 
usually  of  the  second  kind ;  i.e.  the  variable  does  not  reach 
the  limit.  The  limiting  values  of  algebraic  expressions  are 
more  frequently  of  the  first  kind. 

E.q.,  the  function  has  the  limit  1  when  x  becomes  zero :  it  has 

the  limit  0  when  x  becomes  infinite.     The  function  sin  x  has  the  limit  0 

when  x  becomes  zero  ;  tan  x  has  the  limit  1  when  x  becomes  y. 

4 

EXERCISES 

1.  Let  xp  (x,  y)  =  Ax  +  By  -\-  C ;  show  that  ij;  (x,  y)  =0,\f/  (y,  —x)=0 
are  the  equations  of  two  perpendicular  lines. 

2.  If  f(x)  =  2  xVl-  x\  show  that  ffain-^  =  sin  x  =ffcoa-\ 

3.  If  4>  (x)  =  ^^,  show  that  <f>(^)-<t>(y)   =  E^LIL. 

4.  K  f{x)  =  log  f^,  show  that  f{x)  +f{y)  =fl^±JL\. 

1  -\-  X  \1  +  xy/ 

5.  Given  /(x)  =  Vl^^,  find  /( VF^^). 

6.  If  f(xy)  =/(x)  +f(y),  prove  that /(I)  =  0. 

7.  Given  f(x  +  y)=f(x)+f(y),  show  that  /(O)  =  0,  and  that 
pf(^x)  =f(px),  p  being  any  positive  integer. 

8.  Using  the  same  notation  as  in  the  last  example,  prove  that 
/(mx)  =  mf(x),  m  being  any  rational  fraction. 

2.  Infinitesimals  and  infinites.  A  variable  that  approaches 
zero  as  a  limit  is  an  infinitesimal .  In  other  words,  an  infini- 
tesimal is  a  variable  that  becomes  smaller  than  any  number 
that  can  be  assigned. 

The  reciprocal  of  an  infinitesimal  is  then  a  variable  that 
becomes  larger  than  any  number  that  can  be  assigned,  and 
is  called  an  infinite  variable. 

E.g.,  the  number  (J)"  is  an  infinitesimal  when  n  is  taken  larger  and 
larger  ;  and  its  reciprocal  2'»  is  an  infinite  variable. 


1-2.]  FUNDAMENTAL  PRINCIPLES  3 

From  the  definitions  of  the  words  "  limit "  and  "  infinitesi- 
mal" the  following  useful  corollaries  are  immediate  inferences. 

Cor.  1.  The  difference  between  a  variable  and  its  limit 
is  an  infinitesimal  variable. 

Cor.  2.  Conversely,  if  the  difference  between  a  constant 
and  a  variable  be  an  infinitesimal,  then  the  constant  is  the 
limit  of  the  variable. 

For  convenience,  the  symbol  =  will  be  used  to  indicate 
that  a  variable  approaches  a  constant  as  a  limit ;  thus  the 
symbolic  form  x  =  a  is  to  he  read  "  the  variable  x  approaches 
the  constant  a  as  a  limit." 

The  special  form  a;  =  oo  is  read  "a?  becomes  infinite." 

The  corollaries  just  mentioned  may  accordingly  be  sym- 
bolically stated  thus : 

1.  li  X  =  a,  then  x  =  a  +  a,  wherein  a  =  0 ; 

2.  li  X  =  a  -\-  a,  and  a  =  0,  then  x  =  a. 

It  will  appear  that  the  chief  use  of  Cor.  1  is  to  convert 
given  limit  relations  into  the  form  of  ordinary  equations, 
so  that  they  may  be  combined  or  transformed  by  the  laws 
governing  the  equality  of  numbers ;  and  then  Cor.  2  will  serve 
to  express  the  result  in  the  original  form  of  a  limit  relation. 

In  all  cases,  whether  a  variable  actually  becomes  equal  to 
its  limit  or  not,  the  important  property  is  that  their  differ- 
ence is  an  infinitesimal.  An  infinitesimal  is  not  necessarily 
in  all  stages  of  its  history  a  small  number.  Its  essence  lies 
in  its  power  of  decreasing  numerically,  having  zero  for  its 
limit,  and  not  in  the  smallness  of  any  of  the  constant  val- 
ues it  may  pass  through.  It  is  frequently  defined  as  an 
"infinitely  small  quantity,"  but  this  expression  should  be 
interpreted  in  the  above  sense.  Thus  a  constant  number, 
however  small  it  may  be,  is  not  an  infinitesimal. 


4  DIFFERENTIAL   CALCULUS  [Ch.  I. 

3.  Fundamental  theorems  concerning  infinitesimals  and 
limits  in  general.  The  following  theorems  are  useful  in 
the  processes  of  the  calculus ;  the  first  three  relate  to  in- 
finitesimals, the  last  four  to  limits  in  general. 

Theorem  1.  The  product  of  an  infinitesimal  a  by  any 
finite  constant  k  is  an  infinitesimal ; 

i.e.^  if  a  =  0, 

then  ka  =  0. 

For,  let  c  be  any  assigned  number.    Then,  by  hypothesis,  a 

can  become  less  than  - ;  hence  ka  can  become  less  than  c,  the 
k 

arbitrary,  assigned  number,  and  is,  therefore,  infinitesimal. 

Theorem  2.  The  algebraic  sum  of  any  finite  number  n 
of  infinitesimals  is  an  infinitesimal ; 

I.e.,  if  a  =  0,  yS=  0,  •••, 

then  a-}-/3-f----  =  0. 

For  the  sum  of  the  n  variables  does  not  at  any  stage 
numerically  exceed  n  times  the  largest  of  them,  but  this 
product  is  an  infinitesimal  by  theorem  1 ;  hence  the  sum 
of  the  n  variables  is  either  an  infinitesimal  or  zero. 

Note.  The  sum  of  an  infinite  number  of  infinitesimals  may  be 
infinitesimal,  finite,  or  infinite,  according  to  circumstances. 

E.ff.f  if  rt  be  a  finite  constant,  and  if  n  be  a  variable  that  becomes 

infinite;  then  — ,  -,  — ,  are  all  infinitesimal  variables;  but 
n2  n   „i 

—  +  ^  +  .••  to  n  terms  =  -,  which  is  infinitesimal, 
n*     n^  n 

while  -  +  -  +  •••  to  n  terms  =  a,  which  is  finite, 

n     n 

and  —  +  —  +  •••  to  n  terms  =  an^,  which  is  infinite. 


3-4.]  FUNDAMENTAL  PRINCIPLES  5 

Theorem  3.  The  product  of  two  or  more  infinitesimals 
is  an  infinitesimal. 

Theorem  4.  If  two  variables  x,  y  be  always  equal,  and 
if  one  of  them,  x^  approach  a  limit  a,  then  the  other  ap- 
proaches the  same  limit. 

Theorem  5.     If  the  sum  of  a  finite  number  of  variables 
be  variable,  then  the  limit  of  their  sum  is  equal  to  tlie  sum 
of  their  limits ; 
I.e.,  lim  (a; +  ?/+•••)=  lim  a: +  liin^  +  •••. 

For,  let  x  =  a^  y  =  ^->  **•• 

Then  a;  =  «  +  «,   y  =  h-\-^,  •-,     [Art.  2,  Cor.  1. 

wherein  «  =  0,    yS  =  0,  •••; 

hence         x  +  y^  .- =(a  +  J +  •••)  +  (« +  ^+  -•); 

but  «4.^4....  =  0,  [Th.  2. 

hence,  by  Art.  2,  Cor.  2, 

lim(x-|- ?/+  •••)=  a  +  h  -\ =lim  a:  +  lim  y  +  •••. 

Theorem  6.  If  the  product  of  a  finite  number  of  varia- 
bles be  variable,  then  the  limit  of  their  product  is  equal  to 
the  product  of  their  limits. 

Theorem  7.  If  the  quotient  of  two  variables  rr,  y  be 
variable,  then  the  limit  of  their  quotient  is  equal  to  the 
quotient  of  their  limits,  provided  these  limits  are  not  both 
infinite,  or  not  both  zero. 

4.  Comparison  of  variables.  Some  of  the  principles  just 
established  will  now  be  used  in  comparing  variables  with 
each  other.  The  relative  importance  of  two  variables  that 
are  approaching  limits  is  measured  by  the  limit  of  their 
ratio. 


6  DIFFERENTIAL   CALCULUS  [Ch.  I. 

Definition.  One  variable  a  is  said  to  be  infinitesimal, 
infinite,  or  finite,  in  comparison  with  another  variable  x  when 
the  limit  of  their  ratio  a  :  a;  is  zero,  infinite,  or  finite. 

In  the  first  two  cases,  the  phrase  "  infinitesimal  or  infinite 
in  comparison  with  "  is  sometimes  replaced  by  the  less  pre- 
cise phrase  "  infinitely  smaller  or  infinitely  larger  than.'* 
In  the  third  case,  the  variables  will  be  said  to  be  of  the  same 
order  of  magnitude. 

The  following  theorem  and  corollary  are  useful  in  com- 
paring two  variables  : 

Theorem  8.  The  limit  of  the  quotient  of  any  two  varia- 
bles a;,  y  is  not  altered  by  adding  to  them  any  two  numbers 
a,  y8,  which  are  respectively  infinitesimal  in  comparison  with 
these  variables; 

i.e., 
provided 

For,  since 

it  follows,  by  theorems  4,  6,  that 

1+- 
,.  x-\-a  ,.  X  ,.  X 
lim 7^  =  lim  —  •  lim r; ; 

y^^       y      1+^ 
y 

but,  by  theorems  7,  5,  and  hypothesis, 
lim 5=1; 

y 

therefore,  lim ^  =  lim  -• 

y+/3         y 


,.      x-\-  a 

y-\-^ 

lim-, 

y 

X          y 

=  0. 

X  ■\-  a      X 

1  +  ^ 

X 

y-\-^~y 

1  ,^' 

4-5.]  FUNDAMENTAL  PRINCIPLES  7 

Cor.  If  the  difference  between  two  variables  rr,  y  be 
infinitesimal  as  to  either,  the  limit  of  their  ratio  is  1,  and 
conversely ; 


i.e.,  if 

^"^-0,  then  ^  =  1. 

y               y 

For,  since 

x-y^x 

y      y 

hence 

-  -  1  =  0,  and  -  =  1.         [Art.  2,  Cor.  2. 

y                y           •■ 

Conversely, 

if 

""^1,  then^~^=0. 

y               y 

For,  by  Art. 

2, 

Cor.  1, 
?      1^0;z...,^-^-0. 

y                   y 

5.  Comparison  of  infinitesimals,  and  of  infinites.  Orders  of 
magnitude.  It  has  already  been  stated  that  any  two  variables 
are  said  to  be  of  the  same  order  of  magnitude  when  the  limit 
of  their  ratio  is  a  finite  number ;  that  is  to  say,  is  neither 
infinite  nor  zero.  In  less  precise  language,  two  variables 
are  of  the  same  order  of  magnitude  when  one  variable  is 
neither  infinitely  larger  nor  infinitely  smaller  than  the  other. 
For  instance,  Tc^  is  of  the  same  order  as  y8  when  h  is  any 
finite  number;  thus  a  finite  multiplier  or  divisor  does  not 
affect  the  order  of  magnitude  of  any  variable,  whether 
infinitesimal,  finite,  or  infinite. 

In  a  problem  involving  infinitesimals,  any  one  of  them,  a, 
may  be  chosen  as  a  standard  of  comparison  as  to  magni- 
tude ;  then  a  is  called  the  principal  infinitesimal  of  the  first 
order,  and  a~^  is  called  the  principal  infinite  of  the  first 
order. 


S  DIFFEBENTIAL   CALCULUS  [Ch.  I. 

To  test  for  the  order  n  of  any  given  infinitesimal  yS  with 
reference  to  the  principal  infinitesimal  a  on  which  it  depends, 
it  is  necessary  to  select  an  exponent  n  such  that 

lim   ^  _  r. 
a  =  0  ^  ""  '*^' 

wherein  ^  is  a  finite  constant,  not  zero. 

When  n  is  negative,  ^  is  infinite  of  order  —  n.  An 
infinitesimal,  or  infinite  of  order  zero,  is  a  finite  number. 

E.g.,  to  find  the  order  of  the  variable  3  x*  —  4  x^,  with  reference  to  x 
as  the  principal  infinitesimal. 

Comparing  with  x^,  x\  x^,  in  succession : 

lim   3x^-4x3  ^   lim  (3 ^2  _  4 ^)  =  q,  not  finite ; 


x  =  0         X 

lim   3  x^  —  4  xS  _    lim 
a;  =  0        r4  a; 


hence  3  x*  —  4  x^  is  an  infinitesimal  of  the  same  order  of  smallness  as  x*; 
that  is,  of  the  third  order. 

The  order  of  largeness  of  an  infinite  variable  can  be  tested 
in  a  similar  way.  For  instance,  if  x  be  taken  as  the  principal 
infinite,  let  it  be  required  to  find  the  order  of  the  variable 
3  2^  _  4  2^.     Comparing  with  a^  and  a^ : 

lim    Sa^-4a^^    lim    (32;_4)=qo; 


Hm    3  2:* -4  2:3        lij^ 


X  =  00  ^  X 


r«(3-g=3; 


hence  3  a:*  —  4  a;^  jg  ^n  infinite  of  the  same  order  of  largeness 
!is  3^,  that  is,  of  tlie  fourth  order. 

The  process  of  finding  the  limit  of  the  ratio  of  two  in- 
finitesimals is  facilitated  by  the  following  principle,  based 


5-6] 


FUNDAMENTAL  PRINCIPLED 


9 


on  theorem  8  of  Art.  4 :  The  limit  of  the  quotient  of  two 
infinitesimals  is  not  altered  by  adding  to  them  (or  subtract- 
ing from  them)  any  two  infinitesimals  of  higher  order, 
respectively. 

lim     3  a;2  +  a:*  _    lim   3  a:2  _  3 
a;  =  0 


E.g., 


4:X^—2X^ 


x  =  04^-2      4 


From  these  definitions  the  following  theorems  are  at  once 
established  : 

Theorem  1.  The  product  of  two  infinitesimals  is  another 
infinitesimal  whose  order  is  the  sum  of  the  orders  of  the 
factors. 

Theorem  2.  The  quotient  of  an  infinitesimal  of  order  m 
by  an  infinitesimal  of  order  n  is  an  infinitesimal  of  order  m—n. 

Theorem  3.  The  order  of  an  infinitesimal  is  not  altered 
by  adding  or  subtracting  another  infinitesimal  of  higher  order. 

6.  Useful  illustrations  of  infinitesimals  of  different  orders. 

lim  sin  0 


Theorem  1. 


-1  .     lim  tan  6 


6 


With  0  as  a  center  and  OA  =  r 
as  radius,  describe  the  circular 
arc  AB.  Let  the  tangent  at  A 
meet  OB  produced  in  D ;  draw 
BO  perpendicular  to  OA,  cutting 
OA  in  O.  Let  the  angle  AOB  =  6 
in  radian  measure, 
then  arc  AB  =  rO, 

CB<^rcAB<AD, 

i.  e, ,  r  sin  6  <r6  <r  tan  6, 

ain6<0<  tan  9, 


1. 


B  D 


OA 


Fia.l. 


by  geometry, 


i.S^' 


10  DIFFERENTIAL   CALCULUlS  [Ch.  I. 

By  dividing  each  member  of  these  inequalities  by  sin  6^ 

n 
sin  u 
but  sec  ^  =  1,  when  ^  =  0, 

v^/^«««  lina      ^         1     „„  J     lim   sin  6      ^ 

hence  ^^^__=1,  and  ^^^-^=1. 

Similarly,  by  dividing  the  inequalities  by  tan  0^ 

tan  0 

hence  li?l_^  =  l,  and  /Pli^=l. 

^  =  ^tan^  ^  =  ^     ^ 

Cor.  1.     The  numbers  6,  sin^,  tan  ^  are  infinitesimals  of 
the  same  order. 

Cor.  2.     The  expressions  sin  6  —  0,  tan  0  —  0  are  infinitesi- 
mal as  to  0. 

Theorem  2.     If   one   angle  0,  of  a  right  triangle,  be  an 
infinitesimal  of  the  first  order,  then  the  hypotenuse  r  and 

the  adjacent  side  x  are  either  both 
finite,  or  they  are  infinitesimals  of 
the  same  order ;  and  the  opposite 
side  1/  is  an  infinitesimal  of  order 
one  higher  than  that  of  r  and  x. 

For  -  =  cos  0,  which   approaches   the  value  1  as  ^  =  0 ; 

T 

hence  x,  r  are  infinitesimals  of  the  same  order ;  which  may 
be  the  order  zero. 

Also  y  =  r  sin  0^ 

and  sin  0  is  of  order  1 ;  therefore  y  is  of  order  one  higher 
than  r,  by  theorem  1,  Art.  5. 


6.]  FUNDAMENTAL  PBINCIPLES  11 

Cor.  In  the  same  case,  if  6  be  of  the  first  order,  and 
if  r  and  x  be  of  the  order  n,  then  the  difference  between 
r  and  x  is  an  infinitesimal  of  order  n  +  2. 

For  7^  —  x^  =  y'^  =  7^  sin^  ^,  r  —  x  = ; 

r  -\-x 

but  the  orders  of  /^,  sin^  ^,  r  +  a:,  are  respectively  2  ^,  2,  w ; 

therefore  by  theorem  2,  Art.  5,  r  —  a;  is  of  order 

2n+2—n=n+± 

Theorem  3.  The  difference  between  the  length  of  an 
infinitesimal  arc  of  a  circle  and  its  chord  is  of  at  least  the 
third  order  when  the  arc  is  the  first  order. 

For,   let   QD  be  the  arc,  and   CB^  DB,  tangents   at   its 
extremities.      Then  by  elementary  geometry 
chord  CD  <  arc  CD  <  BB  +  BC. 

Let  the  angle  B  OD  =  ^  be  taken  as  the  principal  infini- 
tesimal.    Then,  since  arc  p 
CD  =  2  rd^  and  r  is  finite, 
shence;arc  CD  is  of  order  1. 

Again,  since  AD  is  of  o 
order    1    (Th.    2),    and 
angle  ADB  =  ^  is  of  or- 
der 1,   hence   DB  is   of  Fig.  3. 
order  1,  and  DB  —  DA  is  of  order  3  (Th.  2,  Cor.);  therefore 
{DB  -{-BC)-  chord   CD  is  of  order  3. 

Hence  arc  CD  —  chord  CD  is  of  order,  at  least,  three. 

Theorem  4.  The  difference  between  the  length  of  any 
infinitesimal  arc  (of  finite  curvature)  and  its  chord,  is  an 
infinitesimal  of,  at  least,  the  third  order. 

Note.  The  curvature  is  said  to  be  finite  when  the  limiting  ratio  of 
the  length  of  a  small  chord  to  the  acute  angle  between  the  tangents  at 
its  extremities  is  finite,  and  not  zero. 


12 


DIFFERENTIAL   CALCULUS 


[Cii.  I. 


li  FQ  be  such  an  arc,  the  chord  PQ  and  the  angle  TSF 
are,  by  hypothesis,  infinitesimals  of  the  same  order.* 

Let  the  angle  TSP  be  the 
principal  infinitesimal.  Then, 
since 

TSF  =  aS'^^  +  RFS, 

it  follows  that  the  greater  of 
the  latter  two  angles,  say  SQR, 
is  of  the  first  order,  while  the 
other  may  be  of  th§  first  or 
a  higher  order.  Also,  the 
greater  of  the  two  segments 
HQ,  FM,  say  the  latter,  is  of 
the  first  order,  while  MQ  may 
be  of  the  first  or  higher  order. 
Again,  by  theorem  2,  QR,  QS  are  of  the  same  order,  and 
FR^  FS  are  of  the  same  order. 


Fig.  4. 


Now         arc  QF  -  chord  QF<QS+SF-  QF, 
i.e.,  <  iQS-  QE)  +  (SF-RF); 

but  since     QS-QR=  QS(1  -  cos  /3)  =  2  QS sin^^. 


feeo 


m. 


and,  similarly. 


AS'P-^P=2AS'Psin2^, 


and,  since  each  of  these  products  is,  at  least,  of  the  third 
order,  hence  arc  QF  —  chord  QF  is  of,  at  least,  the  third 
order. 


*  If  TSiT  were  of  higher  order  than  P^,  tho  curvature  would  be  zero ; 
If  of  lower  order,  the  curvature  would  be  infinite  ;  the  former  is  the  case  at 
an  inflexion,  the  latter  at  a  cusp. 


6-7.]  FUNDAMENTAL  PRINCIPLES  13 

EXERCISES 

1.  Let  ABC  be  a  triangle  having  a  right  angle  at  C;  draw  CD  per- 
pendicular to  AB,  DE  perpendicular  to  CB,  EF  perpendicular  to  DB, 
FG  perpendicular  to  EB  ;  let  the  angle  BA  C  be  an  infinitesimal  of  the 
first  order,  AB  remaining  finite.     Prove  that : 

CD,  CB  are  of  order  1 ; 

DB,  DE  are  of  order  2  ; 

EB,  EF,  (CB  -  CD)  are  of  order  3  ; 

FB,  FG,  (DB  -  DE)  are  of  order  4. 

2.  Of  what  order  is  the  area  of  the  triangle  ^5C?  BCD'}   CDE'i 

3.  A  straight  line,  of  constant  length,  slides  between  two  rectangular 
straight  lines,  CAA',  CB'B.  Let  AB,  A'B'  be  tw^o  positions  of  the  line. 
Show  that,  in  the  limit,  when  the  two  positions  coincide, 

AAf  ^  CB 
BB'       CA 

7.  Continuity  of  functions.  When  an  independent  variable 
x^  in  passing  from  a  to  5,  passes  through  every  intermediate 
value,  it  is  called  continuous. 

A  function  f(x)  of  an  independent  variable  x  is  said  to 
be  continuous  at  any  value  x^  when  f(x^  is  finite,  real,  and 
determinate,  and  such  that  in  whatever  way  x  approach  rcp 

in  which  f(x-^  is  independent  of  the  law  of  approach. 

From  the  definition  of  a  limit  it  follows  that  corresponding 
to  a  small  increment  of  the  variable  the  increment  of  the 
function  is  also  small,  and  that  corresponding  to  any  number 
€,  previously  assigned,  another  number  h  can  be  determined, 
such  that  when  h  remains  numerically  less  than  3  the 
difference  ^,        ,^      ^,     - 

is  numerically  less  than  e. 


14  DIFFERENTIAL   CALCULUS  [Ch.  I. 

E.g.,  the  function  f(x)  =  x^  +  3  x -\- 2 

is  continuous  at  the  value  a:  =  1. 

/(1)=6,  /(l  +  A)=6  +  5A  +  *« 
/(I  +  h)-f(l)=  bh  +  h^=h(_5  +  h). 

If  the  difference  /(I  +  h)  —  f(l)  is  to  be  less  than,  say,  unr^rnnf*  i*  Js 
only  necessary  that 

I  ^ 1 

(5  +  ^)1000000* 

If  8  =  Ttnihinrsi  ^^^^  ^^^  every  value  of  h  such  that 

it  is  evident  that /(I  +  ^)  — /(I)  is  less  than  nnrixnn)' 

When  a  function  is  continuous  at  every  value  of  x  within 
the  interval  from  a  to  5,  it  is  said  to  be  continuous  within 
that  interval. 

When  a  value  x^  exists  at  which  any  one  of  the  preceding 
conditions  is  not  fulfilled  for  a  given  function  (f>(x)^  the 
function  is  said  to  be  discontinuous  at  a;=  x^ 

E.g.,  the  function  may  become  infinite,  as     ^^   ,  when  x  =  2; 


the  function  may  be  imaginary,  as  Vd  —  x^,  when  x^  >  9 ; 
the  function  may  be  indeterminate,  as  sin  -,  when  a:  =  0 ; 

X 

finally,  the  value  of  the  function  may  depend  upon  the  manner  in 
which  the  variable  approaches  the  value  x^,  as  in  the  function 


A^)= 


1 

2-3*. 

V 

1-3^ 


when  X  =  +  A,  /(x)=  1 ;  when  x  =  -  A,  /(-  A)  =  2  as  A  =  0. 

A  continuous  function  actually  attains  its  limit  for  any 
value  of  the  variable  within  the  region  of  continuity,  and 
the  variable  may  be  substituted  directly. 


7-8.]  FUNDAMENTAL  PRINCIPLES  15 

It  may  be  shown  as  on  p.  14  that  any  polynomial 

aaf^  +  ^2:""^  H [w  a  positive  integer. 

is  continuous  for  every  finite  value  of  x. 

The  ordinary  functions  involving  radicals  and  ratios  are 
continuous  only  for  certain  intervals. 

The  trigonometric  functions  sin  x  and  cos  x  are  continuous 
for  all  real  finite  values  of  x ;  the  other  trigonometric  func- 
tions are  rationally  expressible  in  terms  of  sine  and  cosine. 

Show  that  tan  x  is  discontinuous  when  x=  ^ir. 

The  exponential  function  a^  and  the  logarithmic  function 
logo;  are  each  continuous,  the  former  for  all  finite  values 
of  a;,  the  latter  for  all  finite  positive  values  of  x  [D.  C,  p.  31]. 

8.  Comparison  of  simultaneous  infinitesimal  increments  of 
two  related  variables.  The  last  few  articles  were  concerned 
with  the  principles  to  be  used  in  comparing  any  two  infini- 
tesimals. In  the  illustrations  given,  the  law  by  which  each 
variable  approached  zero  was  assigned,  or  else  the  two  vari- 
ables were  connected  by  a  fixed  relation ;  and  the  object  was 
to  find  the  limit  of  their  ratio.  The  value  of  this  limit  gave 
the  relative  importance  of  the  infinitesimals. 

In  the  present  article  the  particular  infinitesimals  com- 
pared are  not  the  principal  variables  x^  y  themselves,  but 
simultaneous  increments  A,  k  of  these  variables,  as  they  start 
out  from  given  values  rr^,  y-^  and  vary  in  an  assigned  manner, 
as  in  the  familiar  instance  of  the  abscissa  and  ordinate  of  a 
given  curve. 

The  variables  x^  y  are  then  to  be  replaced  by  their  equiva- 
lents x^  -i-h,  y^-\-  Jc^  in  which  the  increments  h,  k  are  them- 
selves variables,  and  can,  if  desired,  be  both  made  to  approach 
zero  as  a  limit ;  for  since  y  is  supposed  to  be  a  continuous 


16  DIFFERENTIAL   CALCULUS  [Ch.  I. 

function  of  x^  its  increment  can  be  made  as  small  as  desired 
by  taking  the  increment  of  x  sufficiently  small. 

The  determination  of  tlie  limit  of  the  ratio  of  k  to  A,  as  h 
approaches  zero,  subject  to  an  assigned  relation  between  x 
and  ?/,  is  the  fundamental  problem  of  the  Differential 
Calculus. 

E.g.^  let  the  relation  be 

let  a:^,  y-^  be  simultaneous  values  of  the  variables  rr,  y ;  and 
when  X  changes  to  the  value  x^  +  A,  let  y  change  to  the 
value  y^  +  h.     Then 

y^j^k  =  Qx^+  A)2  =  x^  +  2x^h^  A2. 
hence  k=2  x-Ji  +  h^. 

This  is  a  relation  connecting  the  increments  h,  k. 

Here  it  is  to  be  observed  that  the  relation  between  the 
infinitesimals  A,  k  is  not  directly  given,  but  has  first  to  be 
derived  from  the  known  relation  between  x  and  y. 

Let  it  next  be  required  to  compare  these  simultaneous 
increments  by  finding  the  limit  of  their  ratio  when  they 
approach  the  limit  zero. 

By  division, 

-=2x^-\-h\ 

hence,  A^O^=^^i- 

This  result  may  be  expressed  in  familiar  language  by 
saying  that  when  x  increases  through  the  value  x^^  then  y 
increases  2  x^  times  as  much  as  x ;  and  thus  when  x  continues 


8.] 


FUNDAMENTAL   PRINCIPLES 


17 


to  increase  uniformly,  1/  increases  more  and  more  rapidly. 
For  instance,  when  x  passes  through  the  value  4,  and  y 
through  the  value  16,  the  limit  of  the  ratio  of  their  incre- 
ments is  8,  and  hence  y  is  changing  8  times  as  fast  as  x ;  but 
when  X  is  passing  through  5,  and  ^  through  25,  the  limit  of 
the  ratio  of  their  increments  is  10,  and  ^  is  changing  10 
times  as  fast  as  x. 

The  following  table  will  numerically  illustrate  the  fact 
that  the  ratio  of  the  infinitesimal  increments  A,  Jc  approaches 
nearer  and  nearer  to  some  definite  limit  when  h  and  Jc  both 
approach  the  limit  zero. 

Let  x^,  the  initial  value  of  x.he  4.  Then  1/^,  the  initial 
value  of  t/,  is  16.  Let  A,  the  increment  of  a;,  be  1.  Then  k, 
the  corresponding  increment  of  ^,  is  found  from^ 

16  +  A:  =  (4  +  A)2; 

thus  Ar=9,  and  y  =  ^'     Next  let  h  be  successively  diminished 
h 

to  the  values  .8,  .6,  .4,  •••.    Then  the  corresponding  values  of 

Jc 
Jc  and  of  -  are  as  shown  in  the  table : 
Ji 


x  =  4  +  A 

y  =  16-\-k 

k 

k 

h 

4+  1 

25 

9 

9 

4 +.8 

23.04 

7.04 

8.8 

4 +  .6 

21.16 

5.16 

8.6 

4 +.4 

19.36 

3.36 

8.4 

4 +.2 

17.64 

1.64 

8.2 

4+.1 

16.81 

.81 

8.1 

4+  .01 

16.0801 

.0801 

8.01 

4  +  A 

16-f  8A  + A2 

8^  +  A2 

8  + A 

18  DIFFERENTIAL   CALCULUS  [Ch.  I. 

Thus  the  ratio  of  corresponding  increments  takes  the 
successive  values  8.8,  8.6,  8.4,  8.2,  8.1,  8.01,  •••,  and  can 
be  brought  as  near  to  8  as  desired  by  taking  h  small  enough. 

As  another  example,  let  the  relation  between  x  and  y  be 

Then  y,^  =  x,^ 

hence,  by  expansion  and  subtraction, 

2  y^^  +  ^2  _  3  x2Ji  +  3  ^.^^2  +  ^8^ 

*(2y,  +  k)=h  (dx^^+SxJi  +  A2), 

k^dx,^-{-  dx,h  +  h^ 
h  2yi+k 

Therefore  lim  f  =  lim  ^^i'+ ^^'^ /  ^',  as  A  =  0,  *  =  0, 

and,  by  Art  4,  theorem  8, 

lim^  =  5^^ 
h      2y, 

The  "  initial  values "  of  x,  y^  have  been  written  with 
subscripts  to  show  that  only  the  increments  A,  k  vary 
during  the  algebraic  process,  and  also  to  emphasize  the 
fact  that  the  limit  of  the  ratio  of  the  simultaneous  incre- 
ments depends  on  the  particular  values  through  which  the 
variables  are  passing,  when  they  are  supposed  to  take 
these  increments.  With  this  understanding  the  subscripts 
will  hereafter  be  omitted.  Moreover,  the  increments  A,  k 
will,  for  greater  distinctness,  be  denoted  by  the  symbols 
Aa:,  Ay,  read  "increment  of  x,"  "increment  of  y." 

Ex.  1.     If  x2  +  2/2=  a\  find  Hm^.     I^t  the  initial  values  of  the 

variables  be  denoted  by  x,  y,  and  let  the  variables  take  the  res}>ective 
increments  Ax,  Ay,  so  that  their  new  values  x  +  Ax,  y  •¥  ^y  shall  still 
satisfy  the  given  relation.     Then 

(x  +  Ax)2  +  (y  +  Ay)2  =  a«. 


8.]  FUNDAMENTAL  PRINCIPLES 

By  expansion,  and  subtraction, 

2  a: .  Aa:  +  (Aa:)^  +  2  y  •  Ay  +  (Ay)2  = 
hence  Ax  (2  or  +  Aa:)  =  -  Ay  (2  y  +  Ay), 

and 

Therefore 


19 


0, 


Ay_      2a;  +  Aa: 
Aa:         2y  +  Ay 

lim    Ay  _         lim     2  a:  +  Aa: 
Aa;  =  OAx~      Ax  =  02^+ Ay~ 

a: 
2^ 

The  negative  sign  indicates  that  when 
Aa:  and  the  ratio  x:y  are  positive,  Ay  is 
negative ;  that  is,  an  increase  in  x  produces 
a  decrease  in  y.  This  may  be  illustrated 
geometrically  by  drawing  the  circle  whose 
equation  is  a;^  -f  y2  ==  a^  (Fig.  5). 

Ex.2.   If  a;2  +  y  =  y2_2a:, 

prove  lim     Ay^2a:  +  2, 

^  Aa:=OAar      2y-l 


Fig.  5. 


Similarly,  when  the  relation  between  x  and  y  is  given  in 
the  explicit  functional  form 


then 
and 

hence 


y -{- Ay  =  cl>(x -{- Ax), 

Ay  =  (i>{x  +  Ax)  —  <i>(x)  =  A<^(a;), 

lim  ^  =  lim  '^(^  +  A^)-</'('^). 

Ax  Ax 


When  the  form  of  <f>  is  given,  the  limit  of  this  ratio  can  be 
evaluated,  and  expressed  as  a  function  of  x.  This  function 
is  then  called  the  derivative  of  the  function  <t>(x)  with 
regard  to  the  independent  variable  x. 

The  formal  definition  of  the  derivative  of  a  function  with 
regard  to  its  variable  is  given  in  the  next  article. 


20  DIFFERENTIAL   CALCULUS  [Ch.  I. 

9.  Definition  of  a  derivative.  If  to  a  variable  a  small 
increment  be  given,  and  if  the  corresponding  increment  of 
a  continuous  function  of  the  variable  be  determined,  then 
the  limit  of  the  ratio  of  the  increment  of  the  function  to 
the  increment  of  the  variable,  when  the  latter  increment 
approaches  the  limit  zero,  is  called  the  derivative  of  the 
function  as  to  the  variable. 

If  (i>{x)  be  a  finite  and  continuous  function  of  a:,  and  Aa; 
a  small  increment  given  to  x^  then  the  derivative  of  <f>(x)  as 
to  X  is 


lim 
Ax  =  0 


(i>(x  +  Aa:)  —  <f>(x')  1  ^    lim    Aj>(a;) 
Aa;  /-Aa:  =  0     ^^ 


It    is    important    to    distinguish    between    lim     ^^  ^  and 

— - — ^^  ^ ;   that  is,  between  the  limit  of  the  ratio  of  two 
lim  Aa; 

infinitesimals  and  the  ratio  of  their  limits.      The  latter  is 
indeterminate  of  the  form  -  and  may  have  any  value ;  but 

the  former  has  usually  a  determinate  value,  as  illustrated  in 
the  examples  of  the  last  article. 


EXERCISES 

1.  Find  the  derivative  of  a;^  —  2  a:  as  to  x. 

2.  Find  the  derivative  of  3  x*  —  4  a:  +  3  as  to  x. 

3.  Find  the  derivative  of  —  as  to  x. 

4x 

3 

4.  Find  the  derivative  of  x*  —  2  H —  as  to  x. 

/[^  10.  Geometrical  illustrations  of  a  derivative.      Some  con- 

ception  of   the  meaning   and   use  of   a  derivative  will   be 
afforded  by  one  or  two  geometrical  illustrations. 

Let  1/  =  <^(a;)  be  a  function  of  x  that  remains  finite  and 
continuous  for  all  values  of  x  between  certain  assigned  con- 


9-10.] 


FUNDAMENTAL  PRINCIPLES 


21 


stants  a  and  b ;  and  let  the  variables  x,  y  be  taken  as  the 
rectangular  coordinates  of  a  moving  point.  Then  the  rela- 
tion between  x  and  y  is  represented  graphically,  within  the 
assigned  bounds  of  continuity,  by  the  curve  whose  equation  is 

y=^^{x). 

Let  (a;^,  y-^^  (x^^  y^  be  the  coordinates  of  two  points  Py, 
Then  it  is  evident  that  the  ratio 


Pg?  oil  this  curve 


-y\ 


Xc  —  x^ 


is  equal  to  tan  a,  wherein  a  is  the  inclination  angle  of  the 
secant  line  P^^'^  to  the  a;-axis.  Let  P^  be  moved  nearer  and 
nearer  to  coincidence  with  P^  so  that  x^  =  x^  y<^  =  y-^.  Then 
the  secant  line  PiP^  approaches  nearer  and  nearer  to  coinci- 
dence with  the  tangent  line  drawn  at  the  point 
the  inclination-angle  a  of 
the  secant  approaches  as 
a  limit  the  inclination 
angle  </>  of  the  tangent 
line. 
Hence, 

tan  a  =  tan  <f>. 


Py  and 


Pi  carg.y*) 


Thus 


X, 

when  X, 


— -^  =  tan  <^, 


2  —  ^v  Vi  —  yy 


—X 


Fig.  6. 


It  may  be  observed  that  if  x^  be  put  directly  equal  to  x^ 
and  y2  to  y^  the  ratio  on  the  left  would,  in  general,  assume 

the  indeterminate  form  -,  as  in  other  cases  of  finding  the 

limit  of  the  ratio  of  two  infinitesimals ;  but  it  has  just  been 
shown  that  the  ratio  of  the  infinitesimals  y^  —  y^,  x^  —  x^  has, 
nevertheless,  a  determinate  limit,  viz.,  tan  <^. 


22  DIFFERENTIAL   CALCULUS  LCh.  1. 

They  are  thus  infinitesimals  of   the   same  order   except 
when  <^  is  0  or  — • 

If  the  differences  x^  —  ajj,  y^  —  y^  be  denoted  by  Aa;,  Ay, 

then  x^  =  x^-\-  Aa:,  3^2  =  ^i  +  ^^  5 

but,  since  y  =  <^(a:), 

it  follows  that         ^j  =  ^^(^i)^  ^2  ~  *^(^2)» 

hence  the  ratio  of  the  simultaneous  increments  may  be 
written  in  the  various  forms 

Aa;      x^  —  x-^  x^  —  x^  Ax 

In  the  last  form  x  is  regarded  as  the  independent  variable 
and  Aa;  as  its  independent  increment ;  the  numerator  is  the 
increment  of  the  function  <^(a;),  caused  by  the  change  of  x 
from  the  value  a;^  to  the  value  x^  +  Aa;.  The  limit  of  this 
ratio,  as  Aa;  =  0,  is  the  value  of  the  derivative  of  the  function 
</>(a;)  when  x  has  the  value  x^.  Here  x^  stands  for  any 
assigned  value  of  x.  Thus  the  derivative  of  any  continuous 
function  <l>(x)  is  another  function  of  x  which  measures  the 
slope  of  the  tangent  to  the  curve  y  =  </)(a;),  drawn  at  the 
point  whose  abscissa  is  x. 

0 
Ex.    Find  the  slope  of  the  tangent  line  to  the  curve  y  =  —  at  the 

point  (1,  2). 

-     lim     -  2(2  X  +  Ax)  ^  _  4. 
Ax  =  0     x\x  +  Ax)a  x« 


10.] 


FUNDAMENTAL  PRINCIPLES 


23 


Hence  tan  ^  =  —  4,  when  a;  =  1 ;  and  the  equation  of  the  tangent  line  at 
the  point  (1,  2)  is  y  -  2  =  -  4(a;  -  1). 

As  another  illustration,  if  the  coordinates  of  P  be  (x,  y)^ 
and  those  of  §,  (a:  + Aa;, «/+ A^),  then 
MN=PR  =  ^x,  and  P>S=EQ=Ai/. 
It  the  area  OAPM  be  denoted  by 
«,  then  z  is  evidently  some  function 
of  the  abscissas;;  also  if  area  OAQN 
be  denoted  by  z  +  Az^  then  the 
area  MNQP  is  Az ;  it  is  the  incre- 
ment taken  by  the  function  z^  when 
X  takes  the  increment  Ax,  But  MNQP  lies  between  the 
rectangles  MR^  MQ ;  hence 


Y 

S 

/ 

R 

" 

A 

X 

0 

M    N 

Fig.  7. 


and 


1/Ax  <Az<(jj  +  Ay')Ax^ 


Therefore,  when  Aa?,  A^,  Az  all  approach  zero, 

v     Az 
lim  —  =  v» 
Ax     ^ 

Thus,  if  the  ordinate  and  the  area  be  each  expressed  as  a 
function  of  the  abscissa,  the  derivative  of  the  area  function 
with  regard  to  the  abscissa  is  equal  to  the  ordinate  function. 

Ex.  If  the  area  included  between  a  curve,  the  axis  of  x^  and  the 
ordinate  whose  abscissa  is  a:,  be  given  by  the  equation 


z  —  a:*, 


find  the  equation  of  the  curve 
Here  y 


lim  ^  -     li™     (a;  +  Aar)8  -  ofi 
Aar      Aic  =  0  Aa: 


lim 


=  ^^^  Q  [3  a;2  +  3  arAa:  +  (Aa:)2]  =  3  a;= 


24  DIFFERENTIAL    CALCULUS  [Ch.  I. 

11.   The  Operation  of  differentiation.     It  has  been  seen  in  a 
number  of  examples  that  when  the  operation  indicated  by 

lim     j>(2;  +  Aa;)  -  <t)(x) 
Ax  =  0  ^x 

is  performed  on  a  given  function  <t>(x)^  the  result  of  the 
operation  is  another  function  of  x.  The  latter  function  may- 
have  properties  similar  to  those  of  (^(a;),  or  it  may  be  of  an 
entirely  different  class. 

The  operation  above  indicated  is  for  brevity  denoted  by 

the  symbol  — ^-^?  and  the  resulting  derivative  function  by 

(^'(x)',  thus, 

#(^)_     lim    A0(a:)_     lim     <^(a:4- Aa;)- <^(a;) 


C?a;     ~  Aa;  =  0      Aa:  Aa:  =  0  ^^ 


<\>\x^. 


The   process   of   performing   this   indicated   operation   is 
called  the  differentiation  of  <t>(x)  with  regard  to  x.     The 

symbol  *  — ,  when  spoken  of  separately,  is  called  the  differ- 
dx 

entiating  operator,  and  expresses  that  any  function  written 
after  it  is  to  be  differentiated  with  regard  to  rr,  just  as  the 
symbol  cos  prefixed  to  <f>Qx^  indicates  that  the  latter  is  to 
have  a  certain  operation  performed  upon  it,  namely,  that 
of  finding  its  cosine. 

The  process  of  differentiating  <^(a;)  consists  of  the  follow- 
ing steps : 

1.  Give  a  small  increment  to  the  variable. 

2.  Compute  the  resulting  increment  of  the  function. 

3.  Divide  the  increment  of  the  function  by  the  increment 
of  the  variable. 

4.  Obtain  the  limit  of  this  quotient  as  the  increment  of 
the  variable  approaches  zero. 

*  This  symbol  is  sometimes  replaced  by  the  single  letter  D. 


11-12.]  FUNDAMENTAL  PRINCIPLES  25 

EXERCISES 
Find  the  derivatives  of  the  following  functions : 

1.  5  3/8  -  2  2/  +  6  as  to  3^.  3.   8  w^  -  4  w  +  10  as  to  2  m. 

2.  7  /2  _  4  ^  -  11  ^8  as  to  «.  sX  4.   2  a;2  _  5  a:  +  6  as  to  x  -  3. 

This  process  will  be  applied  in  the  next  chapter  to  all  the 
classes  of  functions  whose  continuity  within  certain  inter- 
vals has  been  pointed  out  in  Art.  7.  It  will  be  found  that 
for  each  of  them  a  derivative  function  exists ;  that  is,  that 

lim  —^ — -  has  a  determinate  and  unique  value,  and  that  the 

curve  1/  =  (t>{x)  has  a  definite  tangent  within  the  range  of 
continuity  of  the  function. 

A  few  curious  functions  have  been  devised,  which  are  continuous  and 
yet  possess  no  definite  derivative ;  but  they  do  not  present  themselves  in 
any  of  the  ordinary  applications  of  the  Calculus.     Again,  there  are  a  few 

functions  for  which  lim      \}  ^  has  a  certain  value  when  Ax  =  0  from 

Aa; 

the  positive  side,  and  a  different  value  when  Aa:  =  0  from  the  negative 
side ;  the  derivative  is  then  said  to  be  non-unique. 

Functions  that  possess  a  unique  derivative  within  an  as- 
signed interval  are  said  to  be  differentiahle  in  that  interval. 

Ex.     Show  that  the  four  steps  of  p.  24  do  not  apply  at  a  discontinuity. 

12.  Increasing  and  decreasing  functions.  A  good  example 
of  the  use  of  the  derivative  is  its  application  to  finding  the 
intervals  of  increasing  or  decreasing  for  a  given  function. 

A  function  is  called  an  increasing  function  if  it  increases 
as  the  variable  increases  and  decreases  as  the  variable  de- 
creases. A  function  is  called  a  decreasing  function  if  it 
decreases  as  the  variable  increases,  and  increases  as  the 
variable  decreases. 

E.g.,  the  function  a;^  +  4  decreases  as  x  increases  from  —  oo  to  0,  but 
it  increases  as  x  increases  from  0  to  +  oo.     Thus  a:'^  +  4  is  a  decreasing 


26 


DIFFERENTIAL   CALCULUS 


[Ch.  I. 


function  while  x  is  negative,  and  an  increasing  function  while  x  is  posi- 
tive.    This  is  well  shown  by  the  locus  of  the  equation  y=x^-\-^  (Fig*  8). 


Fig.  8. 

Again,  the  form  of  the  curve  y  =  -  shows  that  -  is  a  decreasing  func- 
tion, as  X  passes  from  —  co  to  0,  and  also  a  decreasing  function,  as  x 
passes  from  0  to  +  oo.  When  x  passes  through  0,  the  function  changes 
discontinuously  from  the  value  —  co  to  the  value  +  oo  (Fig.  9). 

Most  functions  are 
increasing  functions 
for  some  values  of  the 
variable,  and  decreas- 
ing functions  for 
others. 


Fia.  10.  ^'9">    v/2  rx-  x^   is  an 

increasing    function    from 
a;  =  0  to  a;  =  r,  and  a  decreasing  function  from  x  =  rio  x=2r  (Fig.  10). 

A  function  is, said  to  be  an  increasing  or  decreasing  func- 
tion in  the  vicinity  of  a  given  value  of  x  according  as  it 
increases  or  decreases  as  x  increases  through  a  small  interval 
including  this  value. 


12-13.] 


FUNDAMENTAL  PRINCIPLES 


27 


13.  Algebraic  test  of  the  intervals  of  increasing  and  de- 
creasing. Let  1/  =  </>(a;)  be  a  function  of  x^  and  let  it  be  real, 
continuous,  and  differentiable  for  all  values  of  x  from  a  to  h. 
Then  by  definition  i/  is  increasing  or  decreasing  at  a  point 
X  =  x^,  according  as 

is  positive  or  negative,  where  Ax  is  a  small  positive  number. 
The  sign  of  this  expression  is  not  changed  if  it  be  divided 
by  Ax,  no  matter  how  small  Ax  may  be ;  hence  0(:?^)  is  an 
increasing  or  a  decreasing  function  at  the  value  x^,  accord- 
ing as 

Ax 


^=  .  lim     f  <^(^i  +  Ax) -  cl>(x{)  I  ^  ^,^^^^ 


dx 


is  positive  or  negative. 

Thus  the  intervals  in  which  (i>(x)  is  an  increasing  function 
are  the  same  as  the  intervals  in  which  <t>'(x)  is  positive. 

Ex.     Find  the  intervals  in  which  the  function 

,      f^{x)  =  2  a:^  -  9  a:2  +  12  a;  -  6 

is  increasing  or  decreasing.     The  derivative  is 

<^'(a:)=  6a:2  -  18a;  +  12  =  6(a:  -  l)(a:  -  2); 

hence,  as  x  passes  from  -co  to  1,  the  derived  function  ^'(x),  is  positive 

and  ^{x)  increases  from  ^(  — go)  to  f^(l), 

i.e.,  from  <f>=  —  ao  to  <^=— 1;  as  x  passes 

from  1  to  2,  <f>'(x)  is  negative,  and  <l>(x) 

decreases  from  ^(1)   to  <^(2),  i.e.,  from 

—  1  to  -  2 ;  and  as  x  passes  from  2  to  +qo, 

<j!)'(x)  is  positive,  and  <f>(x)  increases  from 

<f}(2)   to    <^(od),  i.e.,  from   —  2    to    +  oo. 

The   locus   of    the    equation    y  =  cf>(x)    is 

shown    in    Fig.   11.       At    points    where 

<l>'(x)=0,  the   function    <^(x)  is  neither 

increasing  nor  decreasing.    At  such  points 

the  tangent  is  parallel  to  the  axis  of  x. 

Thus  in  this  illustration,  at  a:  =  1,  x  =  2, 

the  tangent  is  parallel  to  the  x-axis.  Fig.  11. 


28  DIFFERENTIAL    CALCULUS  [Ch.  I. 

EXERCISES 

1.  Find  the  intervals  of  increasing  and  decreasing  for  the  function 

^{x)  =  a;3  +  2  a;2  +  a:  -  4. 

Here  <^'(a;)  =  3  a:2  +  4  x  +  1  =  (3  a:  +  1)  (a:  +  1). 

The  function  increases  from  x  =  —co  to  x  =  —  1;  decreases  from  x  =  —  1 
to  a;  =  —  I ;  increases  from  a;  =  — ^toa;  =  oo. 

2.  Find  the  intervals  of  increasing  and  decreasing  for  the  function 

y  =  a;8  -  2  a;2  +  a:  -  4, 
and  show  where  the  curve  is  parallel  to  the  a;-axis. 

3.  At  how  many  points  can  the  slope  of  the  tangent  to  the  curve 

3/  =  2a;8-3a:2+l 
be  1  ?    -  1  ?    Find  the  points. 

4.  Compute  the  angle  at  which  the  following  curves  intersect : 

3^  =  3  a:2  -  1,      y  =  2  x^  +  3. 

14.  Differentiation  of  a  function  of  a  function.  Suppose 
that  y,  instead  of  being  given  directly  as  a  function  of  a;, 
is  expressed  as  a  function  of  another  variable  w,  which  is 
itself  expressed  as  a  function  of  x.  Let  it  be  required  to 
find  the  derivative  of  y  with  regard  to  the  independent 
variable  x. 

Let  y=f(u)'>  in  which  tt  is  a  function  of  x.  When  x 
changes  to  the  value  x  +  Aa;,  let  u  and  y^  under  the  given 
relations,  change  to  the  values  w-f-Aw,  y  +  Ay.     Then 

A^  _  A^    Aw  _  f(u-\-Au)  —  f(u')  ^  Aw . 

Ax      A?/     Ax  An  Ax 

hence,  equating  limits, 

^y  _  ^U    ^^^  _  df(u)  ^  du 
dx      du     dx         du        dx 


13-14.]  FUNDAMENTAL   PRINCIPLES  29 

This  result  may  be  stated  as  follows; 

The  derivative  of  a  function  of  u  with  regard  to  x  is  equal  to 
the  product  of  the  derivative  of  the  function  with  regard  to  u, 
and  the  derivative  of  u  with  regard  to  x. 

EXERCISES 

1.  Given  y-'^u^-l,  w  =  3a;2+4:;   find  ^ 

dx 

dy      ^       du      ^ 
du  dx 

2.  Given  y  =  8  m2  _  4  m  +  5,  m  =  2  a:^  -  5 ;   find  $^ 

dx 

3.  Given  ^  =  i,  m  =  5  x2  -  2  a:  +  4;   find  $• 

u  dx 

4.  Given,  =  3„^  +  ^,„=^+|;  find  g. 


CHAPTER  II 
DIFFERENTIATION  OF  THE  ELEMENTARY  FORMS 

In  recent  articles,  the  meaning  of  the  symbol  -^  was  ex- 
plained and  illustrated;  and  a  method  of  expressing  its 
value,  as  a  function  x^  was  exemplified,  in  cases  in  wliich  y 
was  a  simple  algebraic  function  of  a;,  by  direct  use  of  the 
definition.  This  method  is  not  always  the  most  convenient 
one  in  the  differentiation  of  more  complicated  functions. 

The  present  chapter  will  be  devoted  to  the  establishment 
of  some  general  rules  of  differentiation  which  will,  in  many 
cases,  save  the  trouble  of  going  back  to  the  definition. 

The  next  five  articles  treat  of  the  differentiation  of  alge- 
braic functions  and  of  algebraic  combinations  of  other  differ- 
entiable  functions. 

15.  Differentiation  of  the  product  of  a  constant  and  a 
variable. 


Let 

y=^cx. 

Then 

^  +  A?/  =  (?(a;  +  Aa;), 

Ay  =  cQc  -f  Aa;)  —  ca:  =  cAr, 

therefore 

t- 

SO 


Ch.  II.  15-16.]  DIFFERENTIATION  OF  ELEMENTARY  FORMS   31 

Cor.  If  y  =  cu^  where  w  is  a  function  of  x^  then,  by 
Art.  14,  ^,     ^ 

dx  doc 

The  derivative  of  the  product  of  a  constant  and  a  variable  is 
equal  to  the  constant  multiplied  hy  the  derivative  of  the  variable. 

16.  Differentiation  of  a  sum. 

Let  y=f(x)  +  <i>(x)  +  'f{x). 

Then    y  +  ^y  =f(x  +  i^x)  +  <i>(x  +  A:r)  +  -^(x  +  Lx), 

Ay  ^fCx+Ax}-f(x}  ^  (f>(x-hAx}-(l)(x} 
Ax  Ax  Ax 

^'\lr(x  +  Ax}-ylr(x') 

Ax 

Therefore,  by  equating  the  limits  of  both  members, 

g  =/'(^)  +  4,'Cx^  +  ^'(x-).      [Art.  3,  Th.  5. 

Cor.  1.  \i  y  =  u  -^  V  +  w^  in  which  u^  v,  w  are  functions 
of  X,  then  ^       ^     \, 

The  derivative  of  the  sum  of  a  finite  number  of  functions  is 
equal  to  the  sum  of  their  derivatives. 

Cor.  2.     \i  y  =  u  ■\-  c^  c  being  a  constant,  then 
y  -\-  Ay  =  w  +  Au  +  c ; 
hence,  Ay  =  Aw, 

and  ■  dy^du 

dx      dx 

The  last  equation  asserts  that  all  functions  which  differ 
from  each  other  only  by  an  additive  constant  have  the  same 
derivative. 


32  DIFFERENTIAL   CALCULUS  [Ch.  II. 

Geometrically,  the  addition  of  a  constant  has  the  effect  of 
moving  the  curve  y  =  u(x)  parallel  to  the  y  axis  ;  this  opera- 
tion will  obviously  not  change  the  slope  at  points  that  have 

the  same  x, 

^  ,-.  dy      du  ,  dc 

trom(2),  3^  =  T-  +  :r» 

dx      dx     dx 

but  from  the  fourth  equation  above, 

dy  _  du^ 
dx      dx* 

dc 
hence,  it  follows  that         — -  =  0. 

dx 

The  derivative  of  a  constant  is  zero. 

If  the  number  of  functions  be  infinite,  theorem  5  of  Art.  3  may  not 
apply ;  that  is,  the  limit  of  the  sum  may  not  be  equal  to  the  sum  of  the 
limits,  and  hence  the  derivative  of  the  sum  may  not  be  equal  to  the  sum 
of  the  derivatives.  Thus  the  derivative  of  an  infinite  series  cannot  always 
be  found  by  differentiating  it  term  by  term. 

17.  Differentiation  of  a  product. 

Let  y=/(a^)</>(^)- 

Then  ^  =  /(y  +  Aa;)</)(a:  +  Aa:)  -/(a;)  j>(a;) 

Ax  Ax 

By  subtracting  and  adding  f(x)^Qx  +  Ax)  in  the  numer- 
ator, this  result  may  be  rearranged  thus : 

Now  let  Ax  approach  zero,  using  Art.  3,  theorems  5,  6, 
and  noting  that  the  first  factor  (\){x  -h  Ax)  approaches  <j)(x) 
since  by  hypothesis  (f>(x)  is  continuous  (Art.  7).     Then 


16-18.]        DIFFERENTIATION  OF  ELEMENTARY  FORMS       33 

Cor.  1.     By  writing  u  =  (^{x)^  v=f(x)^    this  result    can 

be  more  concisely  written, 

d{uv)  _     dv        du  ^gv 

doc    ~     doc        doc  ^ 

The  derivative  of  the  'product  of  two  functions  is  equal  to  the 
sum  of  the  products  of  the  first  factor  by  the  derivative  of  the 
second^  and  the  second  factor  hy  the  derivative  of  the  first. 

This  rule  for  differentiating  a  product  of  two  functions 
may  be  stated  thus  :  Differentiate  the  product,  treating  the 
first  factor  as  constant,  then  treating  the  second  factor  as 
constant,  and  add  the  two  results. 

Cor.  2.  To  find  the  derivative  of  the  product  of  three 
functions  uvw. 

Let  y  =  uvw. 


/    dv 

=  w[  u—- 

\    dx 


du\  ,        dw 
dxj  dx 


The  result  may  be  written  in  the  form 

dCuvw)  dw  ,         du  ,         dv  ... 

By  application  of  the  same  process  to  the  product  of 
4,  5,  '",  n  functions,  the  following  rule  is  at  once  deduced: 

The  derivative  of  the  product  of  any  finite  number  of  factors 
is  equal  to  the  sum  of  the  products  obtained  by  multiplying  the 
derivative  of  each  factor  by  all  the  other  factors. 

18.  Differentiation  of  a  quotient. 


Let  y 


*(^) 


Then  y  +  Ay=lp^, 


84  DIFFERENTIAL   CALCULUS  [Ch.  11. 

fjx^^x-)  fix-) 
Ay  _  <l>(x-{-Ax)  <t>(x) 
Ax"  Ax 

^  j^jx^fCx  +  Ax)  ^f(x)(t>(x  H-  Ax-) 
Ax(t)(x)(t)(x-\-Ax) 
By  subtracting  and  adding  (\)(x)f{x)  in  the  numerator, 
this  expression  may  be  written 

...  [f(x-\-Ax)-f(ix)\  l^(x+Ax')-<t>{x)\ 

Ax  <t){x)<j)(x -{- Ax) 

Hence,  by  equating  limits, 

dy      <l>(x)f{x)  -f(ix)<i>'(x)  p.        q   Tha   6   7 

di [f(x)Y "-Art.  d,  1  hs.  b,  7. 

This  result  may  be  written  in  the  briefer  form 

du        dv 
d  /  '*  \  _    dx        dx  i-gN 

dQc\v)~         v^         ' 

The  derivative  of  a  fraction^  the  quotient  of  two  functions,  is 
equal  to  the  denominator  multiplied  hy  the  derivative  of  the 
numerator  minus  the  numerator  multiplied  hy  the  derivative 
of  the  denominator,  divided  hy  the  square  of  the  denominator, 

19.  Differentiation  of  a  commensurable  power  of  a  function. 
Let  y  =  w^  in  which  w  is  a  function  of  x.     Then  there  are 
three  cases  to  consider. 

1.  w  a  positive  integer. 

2.  n  a  negative  integer. 

8.    n  a  commensurable  fraction. 

1.    w  a  positive  integer. 

This  is  a  particular  case  of  (4),  the  factors  w,  v,  w,  ••.  all 
being  equal.     Thus 

dx  dx 


18-19.]       DIFFERENTIATION   OF  ELEMENTARY  FORMS       35 


2.    wa 

negative 

integer. 

Let  w  = 

=  -m, 

in 

which  m\8  a.  positive  integer. 

Then 

y=„.=,*-»=i, 

and 

dx 

hence 

dx              dx 

3.    w  a  commensurable  fraction. 

P 
Let  w=— ,  where  jt?,  q  are  both  integers,  which  may  be 

either  positive  or  negative. 

p 
Then  y  =  u''  =  'uP\ 

hence  y^  =  u^^ 


(ia;     ^         dx 
Solving  for  the  required  derivative, 

dx      q  dx^ 

hence  ^  =  nw"  -^  ~  (6) 

doc  dx 

The  derivative  of  any  commensurable  power  of  a  function  is 
equal  to  the  exponent  of  the  power  multiplied  hy  the  power  with 
itB  exponent  diminished  by  unity,  multiplied  by  the  derivative 

of  the  function. 

•'■ 

*  If  two  functions  be  identical,  their  derivatives  are  identical. 


36  DIFFERENTIAL   CALCULUS  [Ch.  II. 

These  theorems  will  be  found  sufficient  for  the  differentia- 
tion of  any  function  that  involves  only  the  operations  of 
addition,  subtraction,  multiplication,  division,  and  involu- 
tion in  which  the  exponent  is  an  integer  or  commensurable 
fraction. 

The  following  examples  will  serve  to  illustrate  the  theo- 
rems, and  will  show  the  combined  application  of  the  general 
forms  (1)  to  (6). 


1.  y  = 


ILLUSTRATIVE  EXAMPLES 
3^'- 2.   ^^^  dy 


X  +  1  ' dx 

dy      (x+l)£(B.-2)-(3x»-2)£(x 

+  1) 

by  (6) 

dx                                (x+l)-^ 

|(^^^-2)=£(^^^>-|(2> 

by  (2) 

=  6x. 

by  (1).  (0) 

i.(x+l)  =  ^  =  l. 

dx                   dx 

by  (2) 

Substitute  these  results  in  the  expression  for  -^'    Then 

dy  _  (x  +  1)6  x  -  (3  x^^  -  2)  ^  3  a:2  +  6  37  +  2 
dx  {x  +  1)2  {x  +  1)2 

2.   m  =  (3«2h-2)V1  +  552;   find—. 

ds 


^=  (3«2  +  2)4  Vl  +  5.2  +  v/TfSTS.  4(3*2+2).     by  (3) 
ds  ds  ds 

4vi  +  5«2=  4(1  +  5*2)* 
€U  ds 

=  |(l  +  5«2)-il(l  +  5«2)         by  (6) 
5» 


Vl  +  5«2 

4(8««4-2)  =  6«.  by  (6) 

ds 


19.]  DIFFEBENriATlON   OF  ELEMENT AUY  FORMS       37 

Substitute  these  values  in  the  expression  for  — •     Then 

ds 


3.  y^VTJ^^+Vljz:^^^  fi^^  rf^. 
VI  +  x^  _  Vl  -  x^  ^a; 

First,  as  a  quotient, 


dy  dx 

dx~  (\/I+^^-V'n^)2 


</ 


( vTT^  +  VT^=^)  —  ( vr+^  -  VI  -  a;2) 

^^"  by  (6) 

(VH-a:2-  Vl-x2)2 


—  (VTT^  +  VI  -  a;2)  =  -^L  VTT^^  +  iL  VI  -  a;2.         by  (2) 
c?a;  dx  dx 

-f  VrT^=  -f  (1  +  :r2)^  =  i(l  +  x'^y-^^il  +  a;2).  by  (6) 

dx  dx  ^  dx 

—  (1  +  x2)  =  2  X.  by  (2)  and  (6> 

dx  • 

Similarly  for  the  other  terms.     Combining  the  results, 


dx       x^   \         Vl  —  x^f 
Ex.  3  may  also  be  worked  by  first  rationalizing  the  denominator. 


EXERCISES 

Find  the  a:-derivatives  of  the  following  functions: 

1.  2/  =  xio.  ^    y  X 

2.  y  =  x~^. 

3.  y  =  cV^.  9.   y 

8/—  

10.    y=(x+  1)  Vx  +  2. 


4. 

V^       3 

5. 

y=^t^. 

6. 

?/  =  (x  +  a)«. 

7. 

y  =  a;«  +  a«. 

11.  y=  ^«+^ 


Va  +  Vx 


--=Vr^- 


38  DIFFERENTIAL   CALCULUS  [Ch.  II. 

X  TQ              3x8+2 

13.  y  = 19.  y= r 

x+Vl-  x^  x{x^-\-\y 

14.  y  =  (2  J  +  x^)  VJ77.  ^20.  y  =  ^x^  +  1)*(4  x^  -  d). 

.„  21.  y=du^-7. 

15.  y=\ [  ' 


^   I  1-  x^ 


16 

17.   y^ 


y/l  _  a;2  >  22.   ?^  =  4  m8  -  6  m2  +  12  M  -  3. 

23.  y=(l-3w2+6M*)(l  +  u2)3. 

24.  y  =  wx. 


X«_4_l  25.     3^  =  m2  ^  3  a;ti2  ^  a;4. 


x«-  1 


y  = 


18.   y  =  -^ 1— .  '"^"^'^ 

(a  +  xl"*     (&  +  x)«  27.   y  =  M%8w. 

28.  Given  (a  +  x)^  =  a^  +  5  a^x  +  10  a^x^  +  10  a^x^  +  5  ax*  +  x^ ;  find 
(a  +  x)*  by  differentiation. 

29.  Show  that  the  slope  of  the  tangent  to  the  curve  y  =  x*  is  never 
negative.     Show  where  the  slope  increases  or  decreases. 

V     30.   Given  b^x^  +  aY  =  a^^^  find  -^ :  (1)  by  differentiating  as  to  x ; 

(2)  by  differentiating  as  to  y;  (3)  by  solving  for  y  and  differentiating 
as  to  X.     Compare  the  results  of  the  three  methods. 

31.   Show  that  form  (1),  p.  31,  is  a  special  case  of  (3). 

^  32.   At  what  point  of  the  curve  y^  =  ax*  is  the  slope  0?     — 1?     +1? 

r  33.   Trace  the  curve  iy  =  x^  +  3  x^  +  x  —  1. 

34.    V  =    '^  "'  -^  "^     and  M  =  5  x2  -  1 ;  find  ^. 
V7  m2  +  5  rfa: 

y  35.    At  what  angle  do  the  curves  y^  =:  12  x  and  y^ -\-  x^  +  Q  x  -  QS  =  0 
intersect  ? 

20.  Elementary  transcendental  functions. 

The  following  functions  are  called  transcendental  func- 
tions : 

Simple  exponential  functions,  consisting  of  a  constant 
number  raised  to  a  power  whose  exponent  is  variable, 
as  4',  a** ; 


19-21.]        DIFFERENTIATION  OF  ELEMENTARY  FORMS       39 

general  exponential  functions,  involving  a  variable  raised 

to  a  power  whose  exponent  is  variable,  as  x^^^ ; 

the  logarithmic  *  functions,  as  log«  x^  log^  u ; 

the  incommensurable  powers  of  a  variable,  as  x^^,  u^ ; 

the  trigonometric  functions,  as  sin  w,  cos  u ; 

the  inverse  trigonometric  functions,  as  sin~^  w,  tan~^  x. 

There  are  still  other  transcendental  functions,  but  they 
will  not  be  considered  in  this  book. 

The  next  four  articles  treat  of  the  logarithmic,  the  two 
exponential  functions,  and  the  incommensurable  power. 

21.  Differentiation  of  loga  oc  and  loga  u. 


Let 

g  =  log„x. 

Then 

g  +  Ag  =  log^(^x  -{-  Ax}, 

^y  _  ^^^a  (^  +  Ax}  -  l0g«  X 

Ax                     Ax 

1  ,       (x^Ax\ 
-AxM      X     } 

For  convenience  writing  h  for  Ax,  and  rearranging. 


Ax 


*  The  more  general  logarithmic  function  log„  w  is  not  classified  separately, 
as  it  can  be  reduced  to  the  quotient  -2E^. 

loga  V 


'    / 

/ 
40  DIFFERENTIAL   CALCULUS  [Ch.  II. 

X 

To  evaluate  the  expression  ( 1  +  -  1    when  A  =  0,  expand  it 

by  the  binomial  theorem,  supposing  -  to  be  a  large  positive 
integer  m. 

The  expansion  may  be  written 

l^W  ~  »»         1-2       ™2+         1.2.3  «t3+     ' 

which  can  be  put  in  the  form 

V^mJ  ^     ^1        2       ^1        2  8       ^ 

1    2 
Now  as  m  becomes  very  lar^e,  the  terms  — ,  — ,  •••  become 
-^       ^  mm 

very  small,  and  when  w«  =  oo  the  series  approaches  the  limit 

1  +  1+  — +  —  +  —  +-. 

2!      3!      4! 

The  numerical  value  of  this  limit  can  be  readily  calculated 
to  any  desired  approximation.  This  number  is  an  important 
constant,  which  is  denoted  by  the  letter  e,  and  is  equal  to 
2.7182818...;  thus 


lim 
m 


^^  (l  +  -X  =  e  =  2.7182818 
=  «>\       mj 


The  number  e  is  known  as  the  natural  or  Naperian  base  ; 
and  logarithms  to  this  base  are  called  natural  or  Naperian 
logarithms.     Natural  logarithms  will  be  written  without  a 

•  This  method  of  obtaining  e  is  rather  too  brief  to  be  rigorous  ;  it  assumes 

that  —  is  a  positive  integer,  but  that  is  equivalent  to  restricting  Aa;  to 
Ax 

approach  zero  in  a  particular  way.    It  also  applies  the  theorems  of  limits  to 

the  sum  and  product  of  an  infinite  number  of  terms.    The  proof  is  completed 

on  p.  316  of  McMahon  and  Snyder's  '*  Differential  Calculus." 


ex- 


21-22.]        DIFFERENTIATION  OF  ELEMENTARY  FORMS       41 

subscript,  as  log  x ;  in  other  bases  a  subscript,  as  in  log^  x, 
will  generally  be  used  to  designate  the  base.  The  logarithm 
of  e  to  any  base  a  is  called  the  modulus  of  the  system  whose 
base  is  a. 

X 

If  the  value,  ;^^o(l+-l  =  e,  be  substituted  in  the 
pression  for  -~,  the  result  is 

di/     1    , 

More  generally,  by  Art.  14, 

£l„„«  =  l.log„e.f.  (7) 

In  the  particular  case  in  which  a  =  e. 

The  derivative  of  the.  logarithm  of  a  function  is  the  product 
of  the  derivative  of  the  function  and  the  modulus  of  the  system 
of  logarithms,  divided  hy  the  function. 

22.  Differentiation  of  the  simple  exponential  function. 

Let  y  =1  a^. 

Then  log  y  =  u  log  a. 

Differentiating  both  members  of  this  identity  as  to  a;, 

,      l^=log«.f^,  byforin(8), 
y  ax  ax 

dy      1  du 

■dx^'^^'^y'Tx' 

therefore  ^a'*  =  log  a  •  a**  •  ~  (9) 

In  the  particular  case  in  which  a  =  «, 

eu  =  e**  •  -j—  (10) 

doc  dx 


42  DIFFERENTIAL    CALCULUS       -  [Ch.  II. 

The  derivative  of  an  exponential  function  with  a  constant  base 
is  equal  to  the  product  of  the  function,  the  natural  logarithm  of 
the  base^  and  the  derivative  of  the  exponent. 

23.  Differentiation  of  the  general  exponential  function. 

Let  y  =  u^^ 

in  which  -m,  v  are  both  functions  of  x. 

Take  the  logarithm  of  both  sides,  and  differentiate.    Then 

logi/  =  v  log  u, 
ydx      dx  udx 


dx        \__  dx     u  dx] ' 


therefore  dx^^  ~  ^^  ^^^  ^  d^^  ^^^^dx  ^^^^ 

The  derivative  of  an  exponential  function  in  which  the  base 
is  also  a  variable  is  obtained  by  first  differentiating^  regarding 
the  base  as  constant^  and  again,  regarding  the  exponent  as 
constant,  and  adding  the  residts. 

In  the  differentiation  of  any  given  function  of  this  form  it 
is  usually  better  not  to  substitute  in  the  formula  directly 
but  to  apply  the  method  just  used  in  deriving  (11),  i.e.,  to 
differentiate  the  logarithm  of  the  function  by  the  preceding 
rules. 

Ex.     y  =  (4  0:2  -  'jy+^^^a^  fi^^  % 

ax 


logy  =  (2  +  Vz2-r5)  log (4 x^ -  7). 

^  =  (4x^-  7)«+^'^«  .^rlog (4x^-7)      8(2  +  VJ^^Ts)-! 
*i^  I     V^a^^  4x»-7       J" 


\  \ 


22-24.]        DIFFERENTIATION  OF  ELEMENTARY  FORMS       43 

24.  Differentiation  of  an  incommensurable  power. 

Let  1/  =  w", 

in  which  n  is  an  incommensurable  constant.     Then 

log  1/  =  n  log  u^ 

\d]£_  n  ^  du 
y  dx      u     dx^ 

du  y     du 

dx  u     ax 

d       n  n-\  du 

dx  dx 

This  has  the  same  form  as  (6),  so  that  the  qualifying  word 
''  commensurable  "  of  Art.  19  can  now  be  omitted. 

EXERCISES 
Find  the  x  derivatives  of  the  following  functions : 

1.  y  =  log(a:+a).  i 

16.   y  =  e^+*. 

2.  y=zlog(ax  +  b). 

3.  y  =  log(4a;2-7a:  +  2).  ^^-   ^  =  i^PgT 

^.   y=  log  ii-^.  18.   y  =  e''(l-  x^). 

1  —  X 

1      l  +  3;2  19.   y  =  - — 

6.  2,=  xlog^.  20.  ./  =  log(6^-.-»). 

7.  2,  =  x-log:r.  21.  2,  =  log  (^  +  .^). 

8.  y  =  3-^  logx-  22.  y  =  x"a»=.  ^ 

9.  y  =  log\/]r^^.  23.  y  =  \og^±^' 

10.  y  =  y/x  -  log  (Va:  +  1).  -i 

11.  y  =  log«(3x2-V2T^).  24.  3/  =  j^- 

12.  3/  =  log,„  {x^  +  7  x).  25.  2/  =  (log  xy. 

13.  ?/  =  log^  a.  26.  2/  =  log  (log  x). 

14.  2/  =  e^^"-  27.  y  =  xf'. 

15.  y  =  e^x+s^  28.  .y  =  a^og==. 


44 


DIFFERENTIAL   CALCULUS 


[Ch.  II. 


The  followiug  functions  can  be  easily  differentiated  by  first  taking 
the  logarithms  of  both  members  of  the  equations. 


29.   v  = 


(^-1) 


(2:-2)^(a:-3)* 


30.   y  =  a;Vl-a:(l  +  x). 


31.  ,  =  £lldL£!l. 

32.  y  =  x\a  +  3  xy{a  -  2  x)^. 

33.  y: 


ovy-'' 


Articles   25-31   will   treat  of  the   differentiation   of   the 
Trigonometric  Functions. 

25.  Differentiation  of  sin  u. 
Let  y  =  sill  u. 

Ay  _  sin  (u  +  Au)  —  sin  2^     A?* 


Then 


Ax  Au  Ax 

_  2  cos  |( 2  1^  +  A?^) sill  I-  Au  ^  Ai* 


A?^ 


Ax 


r       .    1   A     \     Sill  J  Au      Au 
.      s=  COS  (^  +  i  At*) r-^ • 

^         ^               lAu        Ax 
But,  when  Au  =  0,  cos  (?*  H-  |  Au)  =  cos  w,  and  —^. =  1 


by  Art.  6 ;  hence,  passing  to  the  limit. 


iAu 


d    .  du 


(12) 


The  derivative  of  the  sine  of  a  function  is  equal  to  the  prod- 
uct of  the  cosine  of  the  function  and  the  derivative  of  the 
function. 

26.   Differentiation  of  cos  u. 

Let  ^  =  cosw  =  sinf  ^  — wj- 

'^'>-|=f/Kl-")=-<l-")rif-4 


d  ,       du 


(18) 


24-28.]       DIFFERENTIATION  OF  ELEMENTARY  FORMS       45 

The  derivative  of  the  cosine  of  a  function  is  equal  to  minus 
the  product  of  the  sine  of  the  function  and  the  derivative  of  the 
function. 

27.  Differentiation  of  tan  u. 

sin  u 


Let  ?/  =  tan  u 


GOSU 


d    ■  .  d 

cos  u  •  — -  sm  u  —  sin  u  •  —  cos  u 

Then  -/-= 2 ^1  (5) 

ax  coa^u  ^   V  ^ 


»       du  ,     .  o       du        du 
aa^  aa:         dx 


caa^u 


(12),  (13) 


that  is,  ^  tan  M  =  sec^  m  ^- •  (14) 

The  derivative  of  the  tangent  of  a  function  is  equal  to  the 
product  of  the  square  of  the  secant  of  the  function  and  the 
derivative  of  the  function. 

28.  Differentiation  of  cot  u. 

Let  y  =  cot  u  = • 

tan  u 

c^a;      tan^tfc     dx  U\n^u  dx'' 

-5-  cot  t*  =  -  csc^  u  -T—  (15) 

The  derivative  of  the  cotangent  of  a  function  is  equal  to  minus 
the  product  of  the  square  of  the  cosecant  of  the  function  and 
the  derivative  of  the  function. 


46  DIFFERENTIAL   CALCULUS  [Ch.  II. 

29.   Differentiation  of  stcu. 

Let  y  =  sec  u  = 


cosu 


rjy.  dy       —1       d  ,   smu  du 

Then  -f-  =  — k-  •  -j-  cos  w  =  H k-  3-, 

dx     cos^u     dx  cos^  udx 

^8ecu  =  tsLnusecu~  (16) 

dx  dx 

The  derivative  of  the  secant  of  a  function  is  equal  to  the 
product  of  the  secant  of  the  function^  the  tangent  of  the  func- 
tion^ and  the  derivative  of  the  function, 

30.   Differentiation  of  cscu, 

1 

Let  y  —  CSC  u 


sin  w 


rp,  ^„  dy       —  1       d    .  COS  u  du 

Then  -f-  =  ^---  •  — sin  w  =  -  -^— -  -—• 

ax      sm^w     ax  mn^udx 

-P-CSCM  =-C8Ct«C0tW^-  (17) 

dx  dx 

The  derivative  of  the  cosecant  of  a  function  is  equal  to  mimis 
the  product  of  the  cosecant  of  the  function^  the  cotangent  of  the 
function^  and  the  derivative  of  the  function. 

31.  Differentiation  of  verst*. 

Let  y  =  vers  w  =  1  —  cos  u. 

Then  -^  =  —  —  cos  t«. 

dx  dx 

^Ter8W  =  8inM^.  (18) 

dx  dx 

The  derivative  of  the  versed-sine  of  a  function  is  equal  to 
the  produ>ct  of  the  sine  of  the  function  and  the  derivative  of 
the  function. 


29-32.]       DIFFERENTIATION  OF  ELEMENTARY  FORMS       47 

EXERCISES 
Find  the  x  derivatives  of  the  following  functions : 

1.  y  =  sin  7  x,                                      18.  y  =  sin  (m  +  li)  cos  (m  —  6)  / 

2.  y  =  cos  5x.                                       19  —  s^^"*  ^^,     ^ 

3.  y  =  sin  x^.                                             *  cos«  wa: 

4.  3^  =  sin  2  ar  cos  a:,                              20.  «/ =  x  +  log  cosf  a;  -  |  j- 

5.  y  —  sin^  x.  i:j ^ 

7.  y=3m2  7i.  22.  j,  =  sin  (sin «). 

a  2/  =  itan»x-tanx.  23.   i,  =  sin^  e"". 

9.  y  =  sin'acosi.  24,  y  =  sin  «-  •  logx. 

10.  2,  =  tanrt+8ec:t.  25.   t,  =  ^^ii^.  ' 


mx 


12.  ,  =  tan(3-5.T.  27.  ,  =  csc» 4 x.   - 

13.  ?/ =  tan^x  —  logrsec^a;).  _o  ,.  o\?!   ^ 
•^                        ^  ^          ^                28.   ?/  =  sec  (4  x  —  3)2. 

^14.    y  =  log  tan(ia:  +  iTr).  or*  «       •> 

^  °         vz      '   5    y  29.   y  =  vers  a;^. 

15.   y  —  log  sin  Vx.  «^  .l    9  .  /-   -*> 

^         °    J  30.   y  =  cot  x^  +  sec  vx.  ^ 

^  16.  y  =  tan  a*.  31.   ?/  =  sin  xy. 

\         «  4  17.  y  =  sinna:  sin«a:.  32.  y  =  tan  (x  +  y). 

•^      32.  Differentiation  of  Sin- li^.  oM/l^>'^>^ 

Let  ^  =  sin~^w. 

Then  sin  y  =  u^ 

and,  by  differentiating  both  members  of  this  identity, 


hence, 


dy     du 
dx      dx 
dy  _     1     du 1 du  ^ 

dx      cos  y  dx      j.  VI  —  sin^^  dx 


6?     .  _i  \        du 

i.e,j  —-sin  ^u=± —  -— ' 

dx  Vl  —  u^  ^^ 

The  ambiguity  of  sign  accords  with  the  fact  that  sin""^  u 
is  a  many- valued  function  of  u,  since,  for  any  value  of  u  be- 


48  DIFFERENTIAL    CALCULUS  [Ch.  II. 

tween  —  1  and  1,  there  is  a  series  of  angles  whose  sine  is  u : 
and,  when  u  receives  an  increase,  some  of  these  angles  in 
crease  and  some  decrease ;  hence,  for  some  of  them, 


du 

is  positive,  and  for  some  negative.  It  will  be  seen  that, 
when  sin~^w  lies  in  the  first  or  fourth  quarter,  it  increases 
with  tfc,  and,  when  in  the  second  or  third,  it  decreases  as  w 
increases.  Hence,  for  the  angles  of  the  first  and  fourth 
quarters, 

-^sin-^^  =  4-         ^        .     ^sin-ii.^4-— 1— ^.     (19) 

In  the  other  quarters  the  minus  sign  is  to  be  used  before 
the  radical. 

33.  Differentiation  of  the  remaining  inverse  trigonoikfitric 
forms. 

The  derivatives  of  the  other  inverse  trigonometric-  func- 
tions can  be  easily  obtained  by  the  method  employed  in  the 
last  article.     The  results  are  as  follows  : 


^  co8-iw   -      -^      f"     (in  1st  and  2d  quarters). 

(20) 

doc                   Vl  _u'i^iio      ^                          ^             ^ 

/.**-"'"   =,  +  «^^:         (in  all  quarters). 

(21) 

^'''^-x'^'u^Z         0"  all  quarters). 

(22) 

i^  s€c-»M    -        *         "J**  (in  1st  and  3d  quarters). 

(23) 

dx                   uVu^^  ^dx   ^                          ^              ^ 

**  CSC  »t«   -       ~*       ^^    (in  1st  and  3d  quarters). 

(24) 

ax                  uVu^  -idx    ^ 

^  vers-i  w%:         *         ^**  rin  1st  and  2d  quarters^ 
dx                   V2u-U'^^^ 

(26) 

The  radicals  in  forms  (20),  (23),  (24),  (25)  receive  the 
opposite  signs  respectively  when  the  angles  are  taken  in  the 
quarters  other  than  those  stated. 


32-34.]        DIFFERENTIATION   OF  ELEMENTARY  FORMS       49 


EXERCISES 

Find  the  a:-derivative  of  each  of  the  following  functions: 
1.   y  =  sin~^  2  x^.  16.   y  =  tan  x  •  tan~^  x. 


I.   y  =  cos- 1 VI  -  x^.  *H\1.  y  =  x  sin-ix. 


;6.   y  = 
Vl7.   v  = 


/  3.    2^  =  sin- 1(3  a:  -  1). 


18.   v  =  e 


tau-'x 


4.    ?/=  sin-i(3x  — 4x8).  -. 

y<  19.  y  =  csc-^ 


/  5.    w=sm-i^ —•  '^^       ^ 

/  .     .,  ^20.   2/ =  sec-i^-±-!^. 

6.   3/  =  vsin-ix.  "^               x^—1 

-  7.  j,  =  tan-'«-.  ^21.  y  =  taD-'^  +  l". 

S.  y  =  cos-i  log  ar.  1  -  Vaa: 

</9.   w  =  sin-i(tana;).  ^^                  ,ex_e-x 

^              ^          '  22.  2/  =  cos-i^ — 


10.   y=sec-i ^ c*-fe-^ 

VI  -  a:2  V  23.  ?/  =  tan-i(n  tana:). 

•^11.    V  =  vers-i 2 x2.  '                  -iz       o    \ 

^  24.  y  =  cos  '(cos  2  a:). 

12.   y  =  tan-if       ^  j»                   y  25.  ?/ =  cos-i(2cosa:). 

/                              0^:2  26.  w  =  tan-i(Vr+^- a:). 


V  14.    w  =  sin-1  Vsin  x. 


.   y  =  sin-'vsin  a:.  '1  + 


i/  27.  y  =  2tan-i    /izi^. 

^1  +  a: 


15.   y  =  tan-i Jln:^^.  28.  ^  =  tan-i?f^^+  tan-i?lzJ 

^  ^1  +  cosa:  .  ^Va  ^y/3 


34.   Table  of  fundamental  forms. 


d(cu)  =c^'  (1) 

doe  dx    » 

#(«  +  t.4t«)=^+^+^-  (2) 

dx  dx     dx     dx 

d(uv)  =u^+v^.  (3) 

dx  dx        dx         ^ 

^{uvw)         =uv^+uw^-\^vw^'         (4) 
dx  dx  dx  dx 

^du_^dv 

d   u  _    dx         dx 

dx  V  ~  «2  ' 


(6) 


#M"  =nw"  1^.  (6) 

dx  dx 


60 


DIFFERENTIAL   CALCULUS 


l""^"" 

_  loga  e  du 
u     dx 

L''^'^ 

^Idu^ 
u  dx 

dx 

dx 

dx 

dx 

dx 

dx 

dx 

da? 

dx 

da? 

f  taut* 
dx 

da? 

dx 

da? 

f  secw 
dx 

=  sec  11  tan  t^  ^• 
da? 

dx 

=  -CSCMCOtl*^. 

dx 

-^versi* 
dx 

=  8int.f*. 
da? 

^sin-ii* 

1        du 

da? 

Vi-u^dx 

dx 

-1      dw 

^tan-i'ti 
da? 

^      1      du 
l^u^dx 

/-cot-it* 
dx 

-1     du 
1  +  1*2  da? 

^  sec^u 

1         dw. 

dx 

i«  Vt*2  -  1  ^^^ 

:^C8C  »w 

-1        du 

dx 

uy/u^  _  1  <*a5 

^yern-^u 

1         du 

dx 

V2n -«€«<*« 

■VVU^' 


.\du 
dx 


[Ch.  n. 

(7) 
(8) 
(») 

.  (10) 
(11) 
(12) 
(13) 
(U) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(28) 
(24) 
(26) 


34.]  DIFFERENTIATION  OF  ELEMENTARY  FORMS         51 

EXERCISES  ON  CHAPTER  II 
Find  the  x  derivatives  of  the  following  functions : 
1.  3,  =  3  :.^  +  5  x'  -  7.  4ig_   3,  =  (^  +  „)  tan->  Ji_  V5S. 

2.^  =  1  +  1-1.  ^^ 

^'     ^"     '  16.  ,  =  eot-l±^^I±^. 

3.   y=(x-{-  5)V^^^.  ^  X 


4.  ^  =  xVcfi-x^.  ^  17.   2/  =  tan'*a:-2tan2a;+log(sec4a;). 

5.  y  =  x  log  sin  x.  ^  ^g.   y  =  ^M^  +  log(l  -  x\ 

1  —  X 


6.  y  =  ^y/a'^-  a:2. 

-^   7.  2/  =  -eV  '  ""'   "^  5  +  3cosa: 

-^8.  3,  =  tan2z,  2  =  tan-i(2x-l).i^20-   2^  =  ^^^  (l^)    -|tan-i^. 

V  9.  y  =  .V«^  n  =  log  sinz.  .  ;  ■g^  •'-  ^  j^g  (^  +  V^2i:^2)+  sec 


19.   v  =  cos-i§^Ll£2^. 
5  +  3  cos  X 

l  +  x\^      1 


10.   y  =  log-.  '22.   y  =  e%  t*  =  logo:.  ^  ..  ^;^ 

1  _  a;2  23.   y  =  log  s^  +  e«,  s  =  sec  a;. 


11 


Vl  +  a;2  24.   x^  +  yS  _  3  ^^-^  _  q. 

12.  y  =  e*  cos  X.  25.   a:2y2  +  ar^  +  ?/8  _  q. 

13.  ?/  =  vers-M-).  26.   a:^  +  a:  =  y  +  3/8. 


4  sin  3.  27.    xy^-\-x^y  =  x  +  y. 

14.    v  =  tan-i-^^i5_£_. 

3  +  5  cos  a;  28.  y  =  sm(2u-  7),  m  =  log  x^. 


tion? 


29.  For  what  values  of  x  is  the  function  —^ — -  an  increasing  f unc-  (l/jl 
n?  a  +  a;  ^^_^ 

/  Vl  +  a:2  —  1\ 

30.  Prove  that  tan-^  ( j  always  increases  with  x. 

.   Show  that  the  a:-derivative  of  tan-^  A/    ~  ^^^  ^  is  not  a  function 

'  1  +  cos  a: 


31 

oix.  '1  + 

32.  Find  at  what  points  of  the  ellipse  —  +  ^  =  1  the  tangent  cuts  off 
equal  intercepts  on  the  axes.  ^ 

33.  Find  the  points  at  which  the  slope  of  the  curve  y  =  tan  x  is  twice 
that  of  the  line  y  =  x. 

34.  Find  the  angle  which  the  curves  y  =  sin  x  and  y  =  cos  x  make 
with  each  other  at  their  point  of  intersection. 


jj^ 


CHAPTER   III 

•  SUCCESSIVE  DIFFERENTIATION 

35.  Definition  of  nth  derivative.  When  a  given  function 
1/  =  <l>(^x)  is  differentiated  witli  regard  to  x  by  tlie  rules  of 
Cliapter  I,  tlien  the  result 

is  a  new  function  of  x  which  may  itself  be  differentiated  by 
the  same  rules.     Thus, 


dx\dxj      dx 


Cpy 

The  left-hand  member  is  usually  abbreviated  to  -y^,  and 
the  right-hand  member  to  <f>"(x)\  that  is, 

Differentiating  again  and  using  a  similar  notation, 

and  so  on  for  any  number  of  differentiations.     Thus  the 

d^y 
symbol   -t4   expresses  that  y   is   to  be  differentiated   with 

regard  to  x,  and  that  the  resulting  derivative  is  then  to  be 

d^y 
differentiated.     Similarly,  ^  indicates  the  performance  of 

52 


Ch.  III.  35.]  SUCCESSIVE  DIFFERENTIATION  53 

the  operation  —  three  times,  T-fyf-^))-     ^^  general,  the 

^"v/        *^  ctx\dx\dx// 

symbol  -j—  means  that  y  is  to  be  differentiated  n  times  in 

succession  with  regard  to  x. 

Ex.  1.    If  y  =  a:^  +  sin  2  x, 

3^  =  4a:8+2cos2a:, 
dx 

g=12x2-48in2ar, 

^  =  24  ar  -  8  cos  2  ar, 

'-^  =  244-16sin2x. 
dx* 

If  an  implicit  equation  between  x  and  «^  be  given  and  the 
derivatives  of  i/  with  regard  to  x  are  required,  it  is  not 
necessary  to  solve  the  equation  for  eithef  variable  before 
performing  the  differentiation. 

Ex.2.   Given  x^-{-y^-\-4:a^xy=0',  find  ^. 

±^(x^  +  y*  +  ^a^xy)  =  0, 

4a:«+4/^  +  4a2a:^  +  4a2^  =  0. 
dx  dx  ^ 


The  last  equation  is  now  to  be  solved  for  -^, 
dy         x^  +  a^y 


dx         y^  +  a^x 
Differentiating  again, 


(1) 


G?a;2~      dx\y^  +  a^xl 


(y^  +  a^x)  J-  (x^  +  a^y)  -  (x^  +  a^y)  -^  (?/«  +  a^x) 
(z/«  +  «2x)  ^3  z2  +  a2  ^)  -  (a:8  +  a^y)  (3  .^^  ^^  +  a^^ 


54  DIFFERENTIAL   CALCULUS  [Ch.  III. 

The  value  of  -^  from  (1)  is  now  to  be  substituted  in  the  last  equa- 
tion, and  the  resulting  expression  simplified.  The  final  form  may  be 
written : 

dhf  _2  a^xy  -  10  a^:i^y^  -  a\x*  +  j/*)-  3  a;V(x*  +  yQ 

In  like  manner  higher  derivatives  may  be  found. 

36.  Expression  for  the  nth  derivative  in  certain  cases.  For 
certain  functions,  a  general  expression  for  the  nth.  derivative 
can  be  readily  obtained  in  terms  of  w. 

Ex.  1.   If  y  =  ^,  then  -^  =  e*,    -fi^  =  e^    ...,    -^-  e', 
^  dx  dx^  dx" 

where  n  is  any  positive  integer.     If  ^  =  e'*",  -r-^  =  a"e'^. 

Ex.  2.   If  y=  sin  x, 

-i^  =  cos  a;  =  sin  (  X  +  -  )> 
dx  \        2/ 

g=cos(..|)  =  sin(..V1, 


dx»  \         2  1 

U  3/=sinax,  — ^  =  a»  sin  (  ax  +  n  -  V 

c/x"  \  2/ 


EXERCISES    ON    CHAPTER    III 

1.  y=3a:*+5x2+3a:-9;  find^.      6.  y  =  e'\ogx',  find  ^. 

2.  y  =  2x2+ 3a: +  5;  find  0  7.   y  =  xMogx;  find  ^ 


3.   y  =  1 ;  find  g.  /  8.   y  =  sec^x ;  find  ^3. 

/    4.   y  =  x«-i;  find  0.  9.  y  =  logsinx;  find  ^3. 

5.  y  =  c* ;  find  -t-|.  10.   y  =  sin  x  cos  x  ;  find  ^. 


35-36.]  SUCCESSIVE  DIFFEUENTIATION  55 

.  /     11.  y  =  ,J1^  ;  find  '%  ^19.  y  =  cos  mx  ;  find  ^. 

12.  y  =  xMoga;2;  find^,-  .    / 20.  ?/=— J— -;  find  ^. 

13.  y  =  sin  a; ;  find  ^,.  "/2I.  «/  =  log  (a  +  x)-;  find  ^. 

14.  y  =  log  («-  +  e--)  ;  find  ^.  22.  y^=2px;   find  ^. 

15.  ^,  =  (x^-3x  +  3).2.;findg.        23.  ^'+ |-I=  1 ;  find  g. 

16.  y  =  xMoga;;  find  ^.  24.  x^  +  z/3  =  3  ax^/ ;  find  ^. 

17.  2/  =  e«* ;  find  |^.  25.  e^+»  =  a:y ;  find  ^. 

18.  V  =  -ir;  find  ^'  26.   «/  =  1  +  a:e»;  find  ^. 
^      a:  -  1 '           rfx**  ^  '  6?a;2 

rf^y         dy 

27.  y  =  e*  sin  a: ;  prove  y|  —  2-p  +  2y  =  0. 

28.  y=:aa:sina;;  prove  a:2 ^  -  2  a: ^  +  (a:^  +  2) 3/ =  0. 

rf2?/ 

29.  y  =  aa;"+i  +  Ja;-*^ ;   prove  x^j~  =  n(n+  1)  z/. 

30.  y=  (sin-i  a:)^;  prove  (1  "  ^^)  ^2  "  ^  ^  =  2. 

31.  y  =  ^;±^;  prove  ^=l-y'-       33.    y  =  a:-^  log  x  ;  find  ^. 

.  32.   2,  =  -r4-T;  find  ^.  -  34.   y  =  1^;  find  ^f- 

^      ix^-l  dx'^  ^       1  +  a;'  c?x« 

35.  y  =  xV;  prove  ^  =  2|^-'^^  +  2... 

36.  w  =  cos^  X  ;  find  -r-^' 

rfv       1  d"^!!  dy^ 

37.  From  the  relation  -^  =  -— -,  prove  that  -^  =  y-r-rg' 


CHAPTER  IV 
EXPANSION  OF  FUNCTIONS 

It  is  sometimes  necessary  to  expand  a  given  function  in  a 
series  of  powers  of  the  independent  variable.  For  instance, 
in  order  to  compute  and  tabulate  the  successive  numerical 
values  of  sin  x  for  different  values  of  x^  it  is  convenient  to 
have  sin  x  developed  in  a  series  of  powers  of  x  with  coeffi- 
cients independent  of  x. 

Simple  cases  of  such  development  have  been  met  with  in 
algebra.     For  example,  by  the  binomial  theorem, 

(«  +  xy  =  a"  +  na^-^x  +  ^^!^  ~  "^^^""'^  +  '" »        (^) 

J.  *  ^ 

and  again,  by  ordinary  division, 

1 


1-x 


=^\+x-\'x'^  +  a^-\-  ....  (2) 


It  is  to  be  observed,  however,  that  the  series  is  a  proper 
representative  of  the  function  only  for  values  of  x  within  a 
certain  interval.  For  instance,  the  identity  in  (1)  holds 
only  for  values  of  x  between  —  a  and  +  a  when  n  is  not  a 
positive  integer ;  and  the  identity  in  (2)  holds  only  for 
iralues  of  X  between  —  1  and  +  1.  In  each  of  these  ex- 
amples, if  a  finite  value  outside  of  the  stated  limits  be  given 
to  a:,  the  sura  of  an  infinite  number  of  terms  of  the  series  will 
be  infinite,  while  the  function  in  the  first  member  will  be 

finite. 

66 


Ch.  IV.  37.]  EXPANSION  OF  FUNCTIONS  57 

37.  Convergence  and  divergence  of  series.*  An  infinite 
series  is  said  to  be  convergent  or  divergent  according  as  the 
sum  of  the  first  n  terms  of  the  series  does  or  does  not 
approach  a  finite  limit  when  n  is  increased  without  limit. 

Those  values  of  x  for  which  a  series  of  powers  of  x  is  con- 
vergent constitute  the  interval  of  convergence  of  the  series. 

For  example,  the  sum  of  the  first  n  terms  of  the  geometric 

series 

*  a  +  ax  -^  aoc^  -\-  ao^  -\-  ••• 

IS  s„  =  -^- ^• 

1  —  X 

First  let  x  be  numerically  less  than  unity.  Then  when  n 
is  taken  sufficiently  large,  the  term  x^  approaches  zero ; 

hence  li"V^,^=^. 

Next  let  X  be  numerically  greater  than  unity.     Then  rr"  be- 
comes infinite  when  n  is  infinite ;  hence,  in  this  case 

Thus  the  given  series  is  convergent  or  divergent  according 
as  X  is  numerically  less  or  greater  than  unity.  The  condition 
for  convergence  may  then  be  written 

-l<a;<l, 

and  the  interval  of  convergence  is  between  —  1  and  +  1. 

Similarly  the  geonietric  series 

1  _  3^  +  93,2  _  27  2:3^...^ 

*  For  an  elementary,  yet  comprehensive  and  rigorous,  treatment  of  this 
subject,  see  Professor  Osgood's  " Introduction  to  Infinite  Series"  (Harvard 
University  Press,  1897). 


58  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

whose  common   ratio  is   —  3  a;,  is  convergent  or  divergent 
according  as  3  a;  is  numerically  less  or  greater  than  unity. 

The  condition  for  convergence  is  —  1  <  3a;<  1,  and  hence 
the  interval  of  convergence  is  between  —  J  and  +  J. 

38.  General  test  for  interval  of  convergence.  The  follow- 
ing summary  of  algebraic  principles  leads  up  to  a  test  that 
is  sufficient  to  find  the  interval  of  convergence  for  a  series  of 
the  most  usual  kind,  that  is,  a  series  consisting  of  positive 
integral  powers  of  a;,  in  which  the  coefficient  of  a;"  is  a  known 
function  of  n, 

1.  If  8„  is  a  variable  that  continually  increases  with  w,  but 
for  all  values  of  n  remains  less  than  some  fixed  number  ^, 
then  «„  approaches  some  definite  limit  not  greater  than  h 
[This  follows  from  the  definition  of  a  limit.] 

2.  If  one  series  of  positive  terms  is  known  to  be  conver- 
gent, and  if  the  terms  of  another  series  be  positive  and  less 
than  the  corresponding  terms  of  the  first  series,  then  the 
latter  series  is  convergent.     [Use  1.] 

3.  If,  after  a  given  term,  the  terms  of  a  series  form  a 
decreasing  geometric  progression,  then  : 

(a)  The  successive  terms  approach  nearer  and  nearer  to 
zero  as  a  limit ; 

(5)  The  sum  of  all  the  terms  approaches  some  fixed  con- 
stant  as  a  limit.      [Use  method  of  last  article.] 

4.  If  the  terms  of  a  series  be  positive,  and  if  after  a  given 
term  the  ratio  of  each  term  to  the  preceding  be  less  than  a 
fixed  proper  fraction,  the  series  is  convergent.    [Use  2  and  3.] 

5.  If  there  be  a  series  A  consisting  of  an  infinite  number 
of  both  positive  and  negative  terms,  and  if  another  series  B, 
obtained  therefrom  by  making  all  the  terms  positive,  is 
known  to  be  convergent,  then  the  series  A  is  convergent. 


37-38.]  EXPANSION   OF  FUNCTIONS  59 

For  the  positive  terms  of  A  must  form  a  convergent  series, 
otherwise  the  series  B  could  not  be  convergent;  similarly 
the  negative  terms  of  A  must  form  a  convergent  series. 
Let  the  sums  of  these  convergent  series  be  u  and  —  v.  Let 
the  first  n  terms  of  series  A  contain  m  positive  terms  and  p 
negative  terms.  Let  2^,  —  T^,  S„  denote  the  sum  of  the 
positive  terms,  the  sum  of  the  negative  terms,  and  the  sum 
of  all  n  terms  respectively.  Then  aS^^  =  2^  —  Tp.  Now 
when  n  approaches  infinity,  m  and  p  also  approach  infinity 
and  hence 

lim    cr   __     lim    y     _     lim    m      {  ^     S-v  -  v 

Therefore  the  series  A  is  convergent. 

Definitions.  The  absolute  value  of  a  real  number  x  is 
its  numerical  value  taken  positively,  and  is  written  \x\.  The 
equation  | a: |  =  \a\  indicates  that  the  absolute  value  of  x  is 
equal  to  the  absolute  value  of  a.  When,  however,  x  and  a 
are  replaced  by  longer  expressions,  it  is  convenient  to  write 
the  relation  in  the  form  a;  |  =  |  a,  in  which  the  symbol  |  =  |  is 
read  "equals  in  absolute  value."  In  like  manner,  the  sym- 
bols I  <  I,  I  >  I  will  be  used  to  indicate  that  the  expression  on 
the  left  has  respectively  a  smaller,  or  larger,  numerical  value 
than  the  one  on  the  right. 

Any  series  of  terms  is  said  to  be  absolutely  or  uncon- 
ditionally convergent  when  the  series  formed  by  their  abso- 
lute values  is  convergent.  When  a  series  is  convergent,  but 
the  series  formed  by  making  each  term  positive  is  not 
convergent,  the  first  series  is  said  to  be  conditionally 
convergent.* 

*The  appropriateness  of  this  terminology  is  due  to  the  fact  that  the  terms  of 
an  absolutely  convergent  series  can  be  rearranged  in  any  way,  without  altering 
the  limit  of  the  sum  of  the  series ;  and  that  this  is  not  true  of  a  conditionally 


60  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

E.g.,  the  series ———  +  —  —•••  is  absolutely  convergent;  but  the 
series  \  —  \  +  \—  •••  is  conditionally  convergent. 

6.  If  there  be  any  series  of  terms  in  which  after  some  fixed 
term  the  ratio  of  each  term  to  the  preceding  is  numerically 
less  than  a  fixed  proper  fraction ;  then, 

(a)  the  successive  terms  of  the  series  approach  nearer 
and  nearer  to  zero  as  a  limit ; 

(J)  the  sum  of  all  the  terms  approaches  some  fixed  con- 
stant as  a  limit ;  and  the  series  is  absolutely  convergent. 
[Use  3,  4,  5.] 

Ex.  1.    Find  the  interval  of  convergence  of  the  series 

l+2.2a:+3.4z2+4.8a:3  4-5.16a:4+.... 

Here  the  nth  term  m„  is  n2^-^x'^-'^,  and  the  (n-f  l)st  term  u„+i  is 
(n  +  l)2"a;",  hence 

Mn+i  ^  (n  +  1)  2»a:"  ^  (n  +  1)2  a: 
M„   -    n2~-ix«  1    ~  n  ' 

therefore  when  n  =  co,  -^^  =  2x. 

It  follows  by  (6)  that  the  series  is  absolutely  convergent  when 
—  1  <  2  X  <  1,  and  that  the  interval  of  convergence  is  between  —  |  and 
+  J.  The  series  is  evidently  not  convergent  when  x  has  either  of  the 
extreme  values. 

Ex.  2.    Find  the  interval  of  convergence  of  the  series 
X         x^  x^ £l_  4. .         (—  1)" 


1.3     3-38     5.36     7-3^  (2n-l)32'-i 

®  w,  '"^2n  +  1  *  32«+i  '  a:2«-i~  2n  +  1  '  3«* 


hence  --^-^  =  :77,»  when  n  =  co 


w«+i  .  ar-* 


convergent  series.    Thus  the  numerical  value  of  the  series -  + ••• 

12     2*     .3'^ 

is   independent  of  the  order  or  grouping  ;    but  the  value  of   the   series 

\  —  \-\-  \  —  \-\-  •'•  can  be  made  equal  to  any  nunjber  whatever  by  suitable 

rearrangement.     [For  a  simple  proof,  see  Osgood,  pp.  43,  44.] 


38-39.]  EXPANSION   OF  FUNCTIONS  61 


thus  the  series  is  absolutely  convergent  when  —  <  1,  i.e.,  when  —  3  <  a:  <  3, 

32 

and  the  interval  of  convergence  is  from  —  3  to  +  3.     The  extreme  values 
of  X,  in  the  present  case,  render  the  series  conditionally  convergent. 

E..  3.    Show  that  the  seHes  i(|)-  |^(|)%  J.4|)^-^(|)V  ... 

has  the  same  interval  of  convergence  as  the  last ;  but  that  the  extreme 
values  of  x  render  the  series  absolutely  convergent. 

39.  Remainder  after  n  terms.  The  last  article  treated 
of  the  interval  of  convergence  of  a  given  series  without 
reference  to  the  question  whether  or  not  it  was  the  develop- 
ment of  any  known  function.  On  the  other  hand,  the  series 
that  present  themselves  in  this  chapter  are  the  developments 
of  given  functions,  and  the  first  question  that  arises  is 
concerning  those  values  of  x  for  which  the  function  is 
equivalent  to  its  development. 

When  a  series  has  such  a  generating  function,  the  differ- 
ence between  the  value  of  the  function  and  the  sum  of  the 
first  n  terms  of  its  development  is  called  the  remainder 
after  n  terms.  Thus  if  fQx')  be  the  function,  S„(^x}  the 
sum  of  the  first  n  terms  of  the  series,  and  i2„(a;)  the 
remainder  obtained  by  subtracting  S^C^")  from  /(a;),  then 

in  which  S„(x'),  Jl„(^x')  are  functions  of  n  as  well  as  of  x, 

"  „'L"'co^»C^)  =  0,    then   J™^^,(:,)=/(^); 

thus  the  limit  of  the  series  S„(^x^  is  the  generating  function 
when  the  limit  of  the  remainder  is  zero.  Frequently  this 
is  a  sufficient  test  for  the  convergence  of  a  series  V"  ^nC^^)- 
If  a  series  proceed  in  integral  powers  of  x—a^  the  pre- 
ceding conditions  are  to  be  modified  by  substituting  x  —  a 
for  X ;  otherwise  each  criterion  is  to  be  applied  as  before. 


62  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

40.   Maclaurin's  expansion  of  a  function  in  a  power-series.* 

It  will  now  be  shown  that  all  the  developments  of  functions 
in  power-series  given  in  algebra  and  trigonometry  are  but 
special  cases  of  one  general  formula  of  expansion. 

It  is  proposed  to  find  a  formula  for  the  expansion,  in 
ascending  positive  integral  powers  of  x  —  a^  of  any  assigned 
function  which,  with  its  successive  derivatives,  is  continuous 
in  the  vicinity  of  the  value  x  =  a. 

The  preliminary  investigation  will  proceed  on  the  hypothe- 
sis that  the  assigned  function  f(x)  has  such  a  development, 
and  that  the  latter  can  be  treated  as  identical*  with  the 
former  for  all  values  of  x  within  a  certain  interval  of  equiva- 
lence that  includes  the  value  x  =  a.  From  this  hypothesis 
the  coefficients  of  the  different  powers  of  a:  —  a  will  be  de- 
termined. It  will  then  remain  to  test  the  validity  of  the 
result  by  finding  the  conditions  that  must  be  fulfilled,  in 
order  that  the  series  so  obtained  may  be  a  proper  representa- 
tion of  the  generating  function. 

Let  the  assumed  identity  be 

f(x)^  A  4-  B(^x  -d)+  C(x  -  ay  -f-  I)(x  -  ay 

+  J57(a;-a)4H-...,     (1) 

in  which  A^  B^  (7,  •••  are  undetermined  coefficients  indepen- 
dent of  X, 

Successive  differentiation  with  regard  to  x  supplies  the 
following  additional  identities,  on  the  hypothesis  that  the 
derivative  of  each  series  can  be  obtained  by  differentiating 
it  term  by  term,  and  that  it  has  some  interval  of  equivalence 
with  its  corresponding  function  : 

•  Named  after  Colin  Maclaurin  (1698-1746),  who  published  it  in  his 
♦'Treatise  on  Fluxions"  (1742)  ;  but  he  distinctly  says  it  was  known  by 
Stirling  (1690-1772),  who  also  published  it  in  his  "  Methodua  Differcntialis  " 
(1730),  and  by  Taylor  (see  Art.  41). 


40.]  EXPANSION  OF  FUNCTIONS  63 

f{x)=B-{-2C(^x-a')+      SB{x-ay+  4UCx-ay  +  '" 

f"(x)=^        2(7  +2^'2D{x-a)  +      4:-^  E(x-aY+-: 

f"'(x)=  3.2i)  +^'2>'2E(x-a)  +- 

If,  now,  the  special  value  a  be  given  to  x^  the  following 
equations  will  be  obtained  : 

/(a)=  A,  /'(«)=  B,  f\a)=^2C,  f\a)=  3  •  2  D,  .... 

Hence, 

^  =/(«),   5  =/'(«),   0  =  ^^,    I>  =  i^,  -. 

Thus  the  coefficients  in  (1)  are  determined,  and  the  re- 
quired development  is 

/(ic)  =  /(a)  +  /'(a)(a5  -  a)  +  ^^  (05  -  a)2  +  '^^^(a5  -  «)» 

+  -+^-^^(^ -«)"  +  -.  (2) 

This  series  is  known  as  Maclaurin's  series,  and  the  theo- 
rem expressed  in  the  formula  is  called  Maclaurin's  theorem. 

Ex.  1.    Expand  log  x  in  powers  oi  x  -  a. 

Here       f{x)  =  \ogx,  f(x)  =  I    r(x)=  -  1,  f"(x)  =  ^  ••., 

Hence,   /(a)=loga,  /(«)=^,  /"(«)=-i:2'  /'"(«)  =  ^-' 

^..)^,)^(-l)-Hn-l)! 
and,  by  (2),  the  required  development  is 

log  X  =  log  a  +  1  (;r  -  a)  -  ^^(2:  -  «)2  +  ^3(0:  -  a)8  -  ... 

+  i — i-i —  (a:  -  a)«  +  .-.. 


64  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

The  condition  for  the  convergence  of  this  series  is 

lim    r  {x-aY^^  .  (^-«)"]|^|i. 
n  =  ooL(n  +  l)a"+' *       na''     J'     '    ' 

I.e.,  • -|<|1, 

a 

x-a\<,a, 

0<x<2a. 

This  series  may  be  called  the  development  of  log  x  in  the  vicinity  of 
X  =  a.     Its  development  in  the  vicinity  of  a:  =  1  has  the  simpler  form 

\ogx  =  x-l-lix-lY  +  \{x-iy-..., 

which  holds  for  values  of  x  between  0  and  2. 

Ex.  2.    Show  that  the  development  of  -  in  powers  of  a:  —  a  is 

-  =  -  -  ^  (^  -  «)  +  -3  (^  -  ay-  \  {X  -  ay  + ..., 

X      a      a-  a^  a^ 

and  that  the  series  is  convergent  from  x  =  0  to  x  =  2a. 
Ex.  3.   Develop  e^  in  powers  of  x  —  2. 
Ex.  4.    Develop  x^  —  2x'^  -{■  ox  —  7  in  powers  of  x  —  1. 
Ex,  5,    Develop  3  ?/2  _  14  ?^  _|_  7  in  powers  of  y  —  3. 

The  expansion  of  a  function  /(a:)  in  a  series  of  ascending 
powers  of  x  can  be  obtained  at  once  from  formula  (2)  by 
giving  a  the  particular  value  zero.     The  series  then  becomes 

fix)  =  /(O)  +  r  (0)  X  +  ^^x^  +  -  +  /^"'^<|)^"  +  ♦•■      (3) 

Ex.  6.  Expand  sin  x  in  powers  of  x,  and  find  the  interval  of  conver- 
gence of  the  series. 

Here       /(x)  =  sin  x,  /(O)  =  0, 

/'(.r)=cosx,  /'(0)=1, 

/"(.r)=-sinx,  /'(0)=0, 

/"(x)  =  -cosx,  /"(0)  =  -l, 

/'v(x)=8inx,  /'^•(0)  =  0, 

/v(2-)  =  cosx,  /^(0)=1, 


40.]  EXPANSION  OF  FUNCTIONS  65 

Hence,  by  (3), 

sin  X  =  0  +  1  .  ar  +  0  .  a;2  -  i-ar3  +  0  .  x*  +  ^x^  -" ; 
'~^r^      3!  j,i        5! 

thus  the  required  development  is  '  , 

,,.,  =  , _|.^+l.,._i.,  +  ...  +  ^^.^-x  +  .... 

To  find  the  interval  of  convergence  of  the  series,  use  the  method  of 
Art.  38.     The  ratio  of  Un+i  to  m„  becomes 

Un  '~'(2n+  1)!  ■  (2n-  1)!      (2n+  l)2n* 

This  ratio  approaches  the  limit  zero,  when  n  becomes  infinite,  however 
large  be  the  constant  value  assigned  to  x.  This  limit  being  less  than 
unity,  the  series  is  convergent  for  any  finite  value  of  x,  and  hence  the 
interval  of  convergence  is  from  —  co  to  +00- 

The  preceding  series  may  be  used  to  compute  the  numerical  value  of 
sinx  for  any  given  value  of  x.     Take,  for  example,  x  =  .5  radians.    Then 

'^^  2.3      2.3.4.5     2.3.4.5.6.7         ' 

=  .5000000 

-  .0208333 
+  .0002604 

-  .0000015 
+  .0000000 


sin  (.5)  =  .4794256... 

Show  that  the  ratio  of  u^  to  1/4  is  i^l^  ;  and  hence  that  the  error  in  stopping 
at  W4  is  numerically  less  than  u^l^h  +  (^i^)^  +  "•]»  ^^at  is,  <^\7U^' 

Ex.  7.   Show  that  the  development  of  cos  x  is 

and  that  the  interval  of  convergence  is  from  —  co  to  +  00. 

Ex.  8.   Develop  the  exponential  functions  a*,  e*. 
Here 
f(x)  =  a^,  f(x)  =  a-loga,  f"(x)=a-(\oga)%   -,    /('')(x)=a*(loga)«; 
hence    /(0)=  1,  /'(0)=  loga,  /"(0)  =  (loga)2,   -,  /f«)(0)  =  (loga)», 

and  a—  1  +  (log  a)  x  +  Q^x^  +  ...+  (M^a;»  +  .... 

At  Til 


66  DIFFERENTIAL   CALCULUS  £Ch.  IV. 

As  a  special  case,  put  a  =  e. 
Then  log  a  =  log  e  =  1, 

and  «.=  i  +  .+|!  +  |!+...^£2+.... 

These  series  are  convergent  for  every  finite  value  of  a?. 

41.  Taylor's  series.  If  a  function  of  the  sum  of  two  num- 
bers a  and  x  be  given,  f(a  +  :r),  it  is  frequently  desirable  to 
expand  the  function  in  powers  of  one  of  them,  say  x. 

In  the  function  f(a  +  a;),  a  is  to  be  regarded  as  constant, 
so  that,  considered  as  a  function  of  a;,  it  may  be  expanded  by 
formula  (3)  of  the  preceding  article.  In  that  formula,  the 
constant  term  in  the  expansion  is  the  value  which  the  func- 
tion has  when  x  is  made  equal  to  zero,  hence  the  first  term 
in  the  expansion  of  f(a  -F  x')  may  be  written  f(a).  In  the 
same  manner  the  coefficients  of  the  successive  powers  of  x 
are  the  corresponding  derivatives  of  f(a  +  x)  as  to  rr,  in 
which  X  is  put  equal  to  zero  after  the  differentiation  has 
been  performed.     The  expansion  may  therefore  be  written 

/(a  +  a^)=/(a)+/'(«)aJ  +  ^^ajn...+^^a5"  +  -. 

This  series,  from  the  name  of  its  discoverer,  is  known  as 
Taylor's  series,  and  the  theorem  expressed  by  the  formula  is 
known  as  Taylor's  theorem. 

Ex.     Expand  sin  (a  +  x)  in  powers  of  x. 

Here  /(o  +  a:)  =  sin  (a  +  x), 

hence  /(a)=sina, 

and  f  (n)  =  cos  a, 


T¥           •    X     .     \       •                            sin  a   n     cos  n  o  , 
Heuce  sin  (a  +  a:)  =  sina  +  cos  a  •  x  — KT~^ TT" 


40-42.]     /  EXPANSION   OF  FUNCTIONS  67 

EXERCISES 

1.   Expand  tan  x  in  powers  of  x. 
/  2.   Compare  the  expansion  of  tan  x  with  the  quotient  derived  by 
dividing-  the  series  for  sin  x  by  that  for  cos  x.  ^     -t/" 

See  Exs.  6  and  7,  Art.  40.  a^.u*J) ^"^^^  % 


i.  3.   Prove  log^  ^       ^^^ 

^    X  6       180 


A^rt.  40.  M..^^^^  ^  Ai 


3     x^ 


4.   Prove  log(a;  +  Vl  +  a;2)^a:--:^+-^^-.... 

^    5.   Prove  log cosa:  =  ---  —  --gj g^-.... 

6.  Expand  by  division,  making  use  of  the  exponential  series. 

7.  Find  the  expansion  of  e*  log  (1  +  x)  to  the  term  involving  a:^  by 
multiplying  together  a  sufficient  number  of  terms  of  the  series  for  e*  and 
for  log  (1  +  x). 


8.  Expand  Vl  —  a;^  in  powers  of  x. 

9.  Expand  ^  "^     in  powers  of  x. 

\  —  x 

10.   Arrange  (3  +  xY  -  5(3  +  xy  +  2(3  +  a;)2  _  (3  +  a:)  -  2  in  powers 


42.  K  necessary  restriction  imposed  upon  the  series  so 
that  it  may  be  a  correct  representative  of  the  generating 
function,  is  that  the  remainder  after  n  terms  may  be  made 
smaller  than  any  given  number  by  taking  n  large  enough. 

Before  deriving  the  general  form  for  this  remainder  it  is 
necessary  to  prove  the  following  theorem. 

Rolle's  theorem.  If  f(x)  and  its  first  derivative  are  con- 
tinuous for  all  values  of  x  between  a  and  6,  and  if /(a),/(5) 
both  vanish,  then  f'(x)  will  vanish  for  some  value  of  x  be- 
tween  a  and  h. 

By  supposition  f(x)  cannot  become  infinite  for  any  value 
of  x^  such  that  a<x<h.  If  f  (x)  does  not  vanish,  it  must 
always  be  positive  or  always  be  negative ;  hence,  f(x)  must 


68  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

continually  increase  or  continually  decrease  as  x  increases 
from  a  to  5  (Art.  13). 

This  is  impossible,  since  by  hypotheses  /(a)  =  0  and 
/(5)  =  0 ;  hence,  at  some  point  x  between  a  and  6,  f(x)  must 
cease  to  increase  and  begin  to  decrease,  or  cease  to  decrease 
and  begin  to  increase. 

This  point  x  is  defined  by  the  equation  f'(x^=  0. 
To  prove  the  same  thing  geometrically,  let  t/  =f(x)  be 

the  equation  of  a  continuous 
curve,  which  crosses  the  ic-axis 
at  distances  a:=a,  a;=6>  from  the 
origin.  Then  at  some  point  P 
between  a  and  h  the  tangent  to 
Pj^  j2  ^^®    curve    is    parallel    to    the 

ic-axis,  since  by  supposition 
there  is  no  discontinuity  in  the  slope  of  the  tangent.  Hence 
at  the  point  P 

g  =/'(..)  =0. 

43.  Form  of  remainder  in  Maclaurin's  series.      Let   the 

remainder  after  n  terms  be  denoted  by  Rni^Xt  «)?  which  is 
a  function  of  x  and  a  as  well  as  of  n.  Since  each  of  the 
succeeding  terms  is  divisible  by  (x  —  a)",  R^  may  be  con- 
veniently written  in  the  form 

i2„ (x,  a)  =  y^^  V  ^  (x,  a), 

n  I 

The  problem  is  now  to  determine  ^(a;,  a)  so  that  the 
relation 

fix)  =/(a)  +/'(«)(:.  -  «)  +  £^  Cx  -  a)«  + ] 

(n  — 1)!  n! 


42-43.]  EXPANSION  OF  FUNCTIONS  69 

may  be  an  algebraic  identity,  in  which  the  right-hand  mem- 
ber contains  only  the  first  n  terms  of  the  series,  with  the 
remainder  after  n  terms.     Thus,  by  transposing, 

fix-)  -fia-)  -fiaXx  -a}-  £^  (x  -  af  -  ... 

-fr^i^-ay-^-i^^ix-ay^O.  (2) 

(»  —  1) !  n: 

Let  a  new  function,  FQi),  be  defined  as  follows : 

J-CZ)  =f{x-)  -fiz)  -fiz)(x  -  2)  -  -f^  (X  -  Zy  -  ... 

(n—l)l  nl 

This  function  F(^z)  vanishes  when^  5ir  as  is  seen  by 
inspection,  and  it  also  vanishes  when'^psa,  since  it  then 
becomes  identical  with  the  left-hand  member  of  (2) ;  hence, 
by  Rolle's  theorem,  its  derivative  F'^z')  vanishes  for  some 
value  of  z  between  x  and  a,  say  Zy     But 

-f"(2)=-/'(^)+/'(2)-/"(^)(a'-2)+/"(^)(^-z)--^J 

in-iy.^''     ^■>      +(„_!)!  ^"^      '■>     ■ 

These  terms  cancel  each  other  in  pairs  except  the  last 
two;  hence 

^'^'^  =  %"-%  ^'^^'''  «)-/'"'-(^)]- 

Since  F'(^z}  vanishes  when  z  =  z^  it  follows  that 

<^(^,«)=/'">(z,).  (4) 

In  this  expression  z^  lies  between  x  and  a,  and  may  thus 
be  represented  by 

z^^a  +  eCx-a-),  AMrj\Ji\j 


~i±-^ 


70  DIFFERENTIAL    CALCULUS  [Ch.  IV. 

where  ^  is  a  positive  proper  fraction.     Hence  from  (4) 

(^(a:,  «)=/(«>[«  +  <? (2: -a)], 

A               T>  r        \     f''^\a  +  e(x-a)'\  ,  .„  * 

and  R^  (x,  a)  =  ^ — > — -i—-^ ^  (x  —  a)".  * 

The  complete  form  of  the  expansion  of  f(x)  is  then 
finc>)=fia)+f'{a)  (a?  -  a)  +  f^  (05  -  a)^  +  ... 

in  which  w  is  any  positive  integer.  The  series  may  be  car- 
ried to  any  desired  number  of  terms  by  increasing  /i,  and  the 
last  term  in  (5)  gives  the  remainder  (or  error)  after  the  first 
n  terms  of  the  series.  The  symbol /^"^  (a  +  ^(2;  — a))  indi- 
cates that  f(x)  is  to  be  differentiated  n  times  with  regard  to 
a;,  and  that  x  is  then  to  be  replaced  hj  a  ■\-6(x  —  a), 

44.  Another  expression  for  the  remainder.     Instead  of  put- 
ting Rf^  (x,  a)  in  the  form 

(x  —  a)"  ,  ,       N 

^^ r-^(f>(x,  a), 

n\ 

it  is  sometimes  convenient  to  write  it 

i2„ (x,  a)  =  (a;  -  a)  i/r  (x,  a). 

Proceeding  as  before,  the  expression  for  F'(z)  will  be 

F'(z)  =  -  /"^^^^^ ,  (X -  g)»-'  +  ir{x,  a), 
(w  - 1) ! 

In  order  for  this  to  vanish  when  z  =  Zj,  it  is  necessary  that 

in  which   z^=a  +  6(x  —  d),  x  —  z^  =  (x  —  a)  (\  —  0). 

*  This  form  of  the  remainder  was  found  by  Lagrange  (1736-1813),  who 
published  it  in  the  M^moires  de  TAcad^mie  des  Scieuces  k  Berlin,  1772. 


43-44.]  EXPANSION  OF  FUNCTIONS  71 

Hence  i/r  (x,  a)  =  /"^^  + 6>(a:- a))  ^^  _  Qy-w^  __  ^y-i^ 
(?^  —  1) ! 

"and         i2„(^,  a)  =  (l-^)«-i£!l£±%=^(^_a)«.* 

(n-1)! 

An  example  of  the  use  of  this  form  of  remainder  is  fur- 
nished by  the  series  for  log  x  in  powers  of  a;  —  a,  when  x  —  a 
is  negative,  and  also  in  the  expansion  of  (a  +  a;)"*. 

1.   Find  the  interval  of  equivalence  for  the  development  of  log  a;  in 
powers  of  a:  —  a,  vv^hen  a  is  a  positive  number. 
Here,  from  Art.  40,  Ex.  1, 

hence         /X.  („  +  ,(._„)),= |_£|r_IlL_, 

and,.,Art.43,  ^•^(^,»^l  =  \;:^^^f^^J  =  ll[^^I^J- 

First  let  a:  —  a  be  positive.  Then  when  it  lies  between  0  and  a  it  is 
numerically  less  than  a  +  0{x  —  a),  since  ^  is  a  positive  proper  fraction  ; 
hence  when  n  =  oc 

r ?Lp^ —  T^  0,  and  i?„  (x,  a)  =  0. 

Again,  when  a;  —  a  is  negative  and  numerically  less  than  a,  the  second 
form  of  the  remainder  must  be  employed.     As  before, 

hence                   i^.(.,  a)|H(l  -  ^)-- ■  ^^^^/^l^j. 
1=  1(1  _  ^)n-i (a-x)- 


._.  r(a  -  x)  -  6(a  —  x)!"*-^  a  —  x 

'~'L      a-d(a-x)      J      'a-e(a-x) 


*  This  form  of  the  remainder  was  found  by  Cauchy  (1789-1857),  and  first 
published  in  his  "Legons  sur  le  calcul  infinitesimal,"  1826. 


72  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

The  factor  within  the  brackets  is  numerically  less  than  1,  hence  the 
(n  —  l)st  power  can  be  made  less  than  any  given  number,  by  taking 
n  large  enough.     This  is  true  for  all  values  of  x  between  0  and  a. 

Therefore,  log  x  and  its  development  in  powers  oi  x  —  a  are  equiva- 
lent within  the  interval  of  convergence  of  the  series,  that  is,  for  all 
values  of  x  between  0  and  2  a. 

Ex.  2.  Show  that  the  development  of  arz  in  positive  powers  oi  x  —  a 
holds  for  all  values  of  x  that  make  the  series  convergent;  that  is,  when 
X  lies  between  0  and  2  a. 

If  the  function  is  expanded  in  powers  of  a;,  the  complete 
form  will  be 

/(^)  =/C0) +/'(0>  +  =^  ;^  +  -  +  ^^  :^-> 

^^-^^  (1) 

for  the  first  form  of  remainder,  and 

/W  =/(0)  +/'(0>  +  ^^'a?  +  -  +  -^^af-^ 

for  the  second  form  of  remainder. 

Similarly,  the  complete  form  of  Taylor's  series  (Art.  41) 
becomes 

/(a  +  ^)=/(a)+/'(a>  +  ^^r'+  -  +^^^-' 
for  the  first  form  of  remainder,  and 


/(a  +  rr)  =/(«)+/'(«> +  =^^  a?  +  -  +^--^^' 


(»-l)! 


^^tS:^^^^-^)"-'-^  W 


for  the  second  form  of  remainder. 


44.]  EXPANSION  OF  FUNCTIONS  73 

Ex.   Expand  (a  +  x)""  in  ascending  powers  of  x,  and  determine  the 
interval  within  which  R^  has  the  limit  zero. 

Here  /(a  +  x)  =  (a  +  a:)"», 

hence  f(x)  =  x»", 

and  f'(x)=  mx^-i,  f"(x)  =  m(m  -  l)a;«-2,  ..., 

/(«)(x)=  m(m  -  1)  •••  (m-n  +  l)x'»-»; 
Aa)=a-^,  f'{a)=ma-^-\  f"{a)=  m{m  -  \)a-^-\  ..., 
f^^\a)  =  7?i(m  —  1)  •••  (m  —  n  +  1)0"^-". 

Therefore  (a  +  x)*"  =  a"*  +  ma'^-^x  +  ^^^^~  ^^  a'"-^^;^  +  ... 

^  r/?(m-l)...(m-n  +  2)  ^«-«+i^n-i  ^  ^^(^^  ^^^ 
(w  -  1)  ! 

in  which,  from  the  first  form  of  remainder, 

R,(x,  a)  =  Mm-\)'"(m-n  +  l)  ^^  ^  Qx^-nx- 

n  ! 

^  m{m  -  1)  -  (m  -  n  +  1)  ^^  ^  Sxyl-^Y^ 
n\  \a  +  dxf 

Consider  the  ratio 

7t„       m  —  n  +  1         a; 


When  m  is  greater  than  -  1,  the  factor  ^~  ^ is  less  than  unity 

for  every  value  of  n  greater  than  I  (m+1),  and  when  x  lies  between  0 
and  a  the  second  factor  is  also  less  than  unity.  Their  product  is  there- 
fore less  than  some  fixed  proper  fraction  k  for  every  such  value  of  n. 

Hence  Bn<f<'Rn~i<  F/^n-s  -  <  ^'^t-  lt>l(m  +  l). 

Since  /?,  is  finite  and  ^'»-«  can  be  made  as  small  as  desired  by  taking 
n  sufficiently  large,  it  follows  that 

H"^   Rn(x,a)  =  0 
n=  CO     "^  '    ^ 

when  m>— 1,   and  0<a:<a. 

When  m  is  not  greater  than  -  1,  let  it  be  replaced  by  —  p,  where  p 
is  equal  to  or  greater  than  1.     The  ratio  R^  to  Rn-i  may  now  be  written 

-  1 


(-i)(^+^)-«-fr/ 


74  DIFFERENTIAL   CALCULUS  [Ch.  IV. 

The  factor  ( — f-  1  j  is  equal  to  or  greater  than  unity,  but  in 

the  latter  case  becomes  more  and  more  nearly  unity  as  n  increases. 

The  factor  r—  is  numerically  less  than  1  for  any  value  of  n  when 

a-\-ex  ^  ^ 

z  is  given  any  value  between  zero  and  a.     The   product  is  therefore 

greater  than  unity  for  values  of  n  less  than  some  finite  number  iV,  but 

less  than  unity  for  values  of  n  greater  than  N.     Then,  in  the  same  way 

as  for  the  preceding  case, 

Ry+2<kR!f+l<kmff, 

in  which  Rjf  is  a  finite  number  and  k  a  proper  fraction.     Hence 

for  every  value  of  m,  provided  x  lies  between  0  and  a. 

Since  the  interval  of  convergence  is,  by  Art.  38,  from  x  =  —  a  to  a:  =  a, 
it  remains  to  examine  the  value  of  Rn{x,  a)  when  x  lies  between  0 
and  —  a.  For  this  purpose  it  is  necessary  to  use  the  second  form  of 
remainder,  which  may  be  written, 

a 
But  when  -  is  negative  and  less  than  1,  the  expression Z-Z.  is  a 

proper  fraction,  hence  its  (n  —  l)st  power  approaches  zero  as  a  limit. 

The  expression  (^  -  1)  •  • -(^  -  ^  +  1)  (A'"'^  is  made  up  of  n  -  1  factors 
(n-l)l  \a/ 

of  the  form  ^  ~     .  -,  each  of  which  is  finite  and,  when  k  is  sufficiently 

k       a 
large,  is  numerically  less  than  1 ;  hence  the  limit  of  the  entire  product 

is  zero.     Therefore 

R„(Xf  a)  =  0,  when  n  =  oo, 

if  X  lies  between  —  a  and  +  a. 

Therefore,  for  values  of  x  within  this  interval  the  function  (a  +  x)*^  and 
its  expansion  are  equivalent. 


Y 

/-^ 

^ 

H^,.-^:^^^''^ 

^ 

M 

X 

0 

N                         R 

44-45.]  EXPANSION   OF  FUNCTIONS  75 

45.   Theorem  of  mean  value.     Let  /(a;)  be  a  continuous 
function  of  x  which  has  a  derivative.     It  can  then  be  repre- 
sented by  the  ordinates  of  a  curve 
whose  equation  is  i/  =f(x). 

In  Fig.  13,  let 

x  =  ON,x-\-h  =  OR, 
f(x)  =  NR,  fix  +  h)=  RK, 
Then  f(x-\-K)  -f(x)  =  MK,  and 

h  HM 

But  at  some  point  S  between  H  and  K  the  tangent  to  the 
curve  is  parallel  to  the  secant  HK.  Since  the  abscissa  of  S 
is  greater  than  x  and  less  than  rr  +  A  it  may  be  represented 
by  a;  +  Oh^  in  which  ^  is  a  positive  number  less  than  unity. 
The  slope  of  the  tangent  at  8  is  then  expressed  by  /'(a;+^A), 
hence 

from  which 

f(x  +  }i)=f(x)  +  hf(x  +  eJi), 

The  theorem  expressed  by  this  formula  is  known  as  the 
theorem  of  mean  value. 

The  theorem  of  mean  value  can  also  be  established  from  Taylor's 
theorem  by  putting  w  =  1  in  equation  (3)  of  Art.  44. 


JiA^^ 


EXERCISES  ON  CHAPTER  IV 

1.  Expand  cos(x  +  It)  in  powers  of  h. 

2.  Expand  tan(x  +  h)  in  powers  of  x. 

JL-  3.   By  expanding  cos(a:  +  h)  in  powers  of  one  of  its  variables,  prove 
the  theorem  cos(a:  +  A )  =  cos  x  cos  k  —  sin  x  sin  h. 

4.  Expand  log  (a;  +  ^)  by  Taylor's  theorem  in  powers  of  h. 

5.  Expand  {x  +  y)^  in  powers  of  y. 


76 


DIFFERENTIAL   CALCULUS 


[Ch.  IV.  45. 


6.   Prove  log(l  +  e*)  =  log  2  +  |  +  ^ 


v^     7.   Prove  log 


2     28     28.41 
1  1 


+  i2. 


x-\      x-l      2{x-iy     3(a;-l)' 


+  i2. 


'V.  8.   Prove  log(l  +  sin  ar)  =  a:  -  ^  +  —  -  ^  +  i2. 

9.  If  f(x)=f(—  x),  prove  that  the  expansion  of  f(x)  in  powers  of  x 
will  contain  only  even  powers  of  x,  as  cosx,  y/l  -  x'^;  if  f(x)=  _y(_x), 
the  expansion  oi  f(x)  will  involve  only  odd  powers  of  Xy  as  sin  a:,  tanar, 

X 

1  +  a:2* 

10.   Show  that  in  the  expansion  for  sin  x  in  powerr?  of  x, 
Ijm 


R^(x)=0. 


M 

1- 


'  *■-  jf/  >i-t 


/..  / 


1? 


TL. 


"',  -/ 


.3.  -2'3 


^V 1^, 


2l, 


CHAPTER   V 
INDETERMINATE  FORMS 

46.  Hitherto  the  values  of  a  given  function  f{x)^  corre- 
sponding to  assigned  values  of  the  variable  x^  have  been 
obtained  by  direct  substitution.  The  function  may,  how- 
ever, involve  the  variable  in  such  a  way  that  for  certain 
values  of  the  latter  the  corresponding  values  of  the  function 
cannot  be  found  by  mere  substitution. 

For  example,  the  function 


sma; 
for  the  value  a;=0,  assumes  the  form  -,  and  the  correspond- 
ing value  of  the  function  is  "thus  not  directly  determined. 
In  such  a  case  the  expression  for  the  function  is  said  to 
assume  an  indeterminate  form  for  the  assigned  value  of  the 
variable. 

The  example  just  given  illustrates  the  indeterminateness 
of  most  frequent  occurrence ;  namely,  that  in  which  the 
given  function  is  the  quotient  of  two  other  functions  that 
vanish  for  the  same  value  of  the  variable. 

Thus  if  /(:,)=^, 

and  if,  when  x  takes  the  special  value  a,  the  functions  4>{x) 
and  ^(x)  both  vanish,  then 

is  indeterminate  in  form,  and  cannot  be  rendered  determinate 
without  further  transformation. 

•    77 


78  DIFFERENTIAL   CALCULUS  [Ch.  V. 

47.  Indeterminate  forms  may  have  determinate  values. 
A  case  has  already  been  noticed  (Art.  9)  in  which  an  ex- 
pression that  assumes  the  form  -  for  a  certain  value  of  its 

variable  takes  a  definite  value,  dependent  upon  the  law  of 
variation  of  the  function  in  the  vicinity  of  the  assigned 
value  of  the  variable. 

As  another  example,  consider  the  function 

_x^  —  a^ 
^ ~~  X—  a' 

If  this  relation  between  x  and  y  be  written  in  the.forms 

y{x  —  a)=  x^  —  a\     (x  —  a)(^y  —  x  —  a)  =  0^ 

it  will  be  seen  that  it  can  be  represented  graphically,  as  in 
the  figure  (Fig.  14),  by  the  pair  of  lines 

/  X—  a  =  0, 

*    y  —  X  —  a  =  0. 

Hence  when  x  has  the  value  of  a  there 

— X     is  an  indefinite  number  of  corresponding 

points  on  the  locus,  all  situated  on  the 

^°-  ^*'  line   x  =  a;    and   accordingly  for  this 

value  of  X  the  function  y  may  have  any  value  whatever,  and 

is  therefore  indeterminate. 

When  X  has  any  value  different  from  a,  the  corresponding 
value  of  y  is  determined  from  the  equation  y  =  x  +  a.  Now, 
of  the  infinite  number  of  different  values  of  y  corresponding 
to  x  =  a,  there  is  one  particular  value  AP  which  is  con- 
tinuous with  the  series  of  values  taken  by  y  when  x  takes 
successive  values  in  the  vicinity  of  x=  a.  This  may  be 
called  the  determinate  value  of  y  when  x  =  a.  It  is  ob- 
tained by  putting  x=  a  in  the  equation  y  =  a:  -{-  a,  and  is 
therefore  y  =  2  a. 


47.]  INDETERMINATE  FORMS  79 

This  result  may  be  stated  without  reference  to  a  locus  as 
follows :  When  a;  =  a,  the  function 

00^  —  a^ 


is  indeterminate,  and  has  an  infinite  number  of  different 
values;  but  among  these  values  there  is  one  determinate 
value  which  is  continuous  with  the  series  of  values  taken  by 
the  function  as  x  increases  through  the  value  a ;  this  deter- 
minate or  singular  value  may  then  be  defined  by 

lim  ^-  CL^ 
X  =  a  X  —  a 

In  evaluating  this  limit  the  infinitesimal  factor  x  —  a  may  be 
removed  from  numerator  and  denominator,  since  this  factor 
is  not  zero,  while  x  is  different  from  a ;  hence  the  determi- 
nate value  of  the  function  is 

lim  X  +  a  _  i)  ^ 

Ex.  1.   Find  the  determinate  value,  when  x  =  1,  of  the  function 

x^+2x^-nx 
Sx^-dx^-  X  +  1' 

which,  at  the  limit,  takes  the  form  — 

0 

This  expression  may  be  written  in  the  form 

(x^+^x)(x-  1) 

(3x2- l)(ar-l)' 

x^  4-  3  x 

which  reduces  to  jr— 5 r^.     When  x  =  1,  this  becomes  #  =  2. 

3  x2  -  1  '  ^ 

Ex.  2.   Evaluate  the  expression 

x^  +  ax^  +  a^x  +  a» 
x^  +  xb^  +  ax^  +  ab^ 
when  X  =  —  a. 


80  DIFFERENTIAL  CALCULUS  [Ch.  V. 

Ex.  3.   Determine  the  value  of 

a;3  _  7  3:2  +  3  a:  +  14 


a;3  +  3  x-^  -  17  a;  +  14 
when  X  =  2. 


Ex.  4.  Evaluate         - — ^ — -  when  x=  0. 
x^ 

(Multiply  both  numerator  and  denominator  by  a  +  Va^  —  r^.) 

48.  Evaluation  by  development.  In  some  cases  the  com- 
mon vanishing  factor  can  be  best  removed  after  expansion 
in  series. 

Ex.  1.   Consider  the  function  mentioned  in  Art.  46, 


When  numerator  and  denominator  are  developed  in  powers  of  x,  the 
expression  becomes 

21      3!  V  2\      S\  I 


X- 

3,3 

3! 

+  • 

•• 

2X  +  |;X3 

+ 

... 

2 

,.-,... 

3.8 

^-31  + 

... 

- 

1 

-i^-' 

which  has  the  determinate  value  2,  when  x  takes  the  value  zero. 
Ex.  2.   As  another  example,  evaluate,  when  a:  =  0,  the  function 


X  —  sin-'a: 


sin^a: 
By  development  it  becomes 


Removing  the  common  factor,  and  then  putting  x  =  0,  the  result  is  J. 


47-49.]  INDETERMINATE  FORMS  81 

In  these  two  examples  the  assigned  value  of  x^  for  which 

*    the  indeterminateness  occurs,  is  zero,  and  the  developments 

are  made  in  powers  of  x.     If  the  assigned  value  of  x  be 

some  other  number,  as  a,  then  the  development  should  be 

made  in  powers  of  x  —  a. 

Ex.  3.   Evaluate,  when  x  =  -,  the  function 

cos  a: 


1  —  sin  X 


By  putting  a:  —  -  =  A,  x  =  --\-  h,  the  expression  becomes 

eo,(|+*)  .    ^        -A  +  ^-...      -1  +  f 

\  2         /  —  sm  A  6  6 


l-sin(|+^)       1 


cos  A       '^_A!4-  ^_A! 

2      24  2     24 


which  becomes  infinite  when  A  =  0,  that  is,  when  a;  =  ^. 

TT  hm       cos  X  ,  ^ 

Hence  „  .  ^ : =  ±  co, 

•*^  —  z  1  —  sin  a: 

according  as  h  approaches  zero  from  the  negative  or  positive  side. 

49.   Evaluation  by  differentiation.     Let  the  given  function 

be  of  the  form  ,;  i^  and  suppose  that /(a)  =0,  </>(a)  =  0. 

It  is  required  to  find  j^l^^^i^. 

As  before,  let/(2j),  <^(a;)  be  developed  in  the  vicinity  of 
x  =  a^  by  expanding  them  in  powers  oi  x  —  a.     Then 

fix)      /(«)+/'(«)(^-«)  +  ^^(*-«)'  +  - 
^^'"^  ~  <^(«)  +  f  (a)(^  -  a)  +  *^  (a;  -  «)2  +  ... 
/'(a)(r.-a)+•^(a.-a)''+••. 


82  DIFFERENTIAL   CALCULUS  [Ch.  V. 

By  dividing  hj  x—a  and  then  letting  a:  =  a,  it  follows 
that 

lim  Ax)  ^f{a) 
^  =  ^<t)(x)      (l>\a)' 

The  functions  /\'a),  <^'(a)  will  in  general  both  be  finite. 
If  /(a)  =  0,  <^'(a)  ^  0,  then  ^  =  0. 

If  /(a)  ^  0,  <^'(a)  =  0,  then  ^^  =  oo. 

If /'(a)  and  <^'(a)  are  both  zero,  the  limiting^  value  of 

f(x) 

\\  ^  is  to  be  obtained  by  carrying  Taylor's  development 

one  term  farther,  removing  the  common  factor  (^x  —  a)^,  and 

then  letting  x  approach  a.     The  result  is    f,^  ^- 

Similarly,  if /(a), /(a), /'(«);  (^(a),  </>'(«),  </>''(«)  all 
vanish,  it  is  proved  in  the  same  manner  that 

lim  f(x}_f"(a-) 

and  so  on,  until  a  result  is  obtained  that  is  not  indeterminate 
in  form. 

Hence  the  rule : 

To  evaluate  an  expression  of  the  form  -,  differentiate  numer- 
ator and  denominator  separately/ ;  substitute  the  critical  value 
of  x  in  their  derivatives^  and  equate  the  qw)tient  of  tJie  deriva- 
tives to  the  indeterminate  form, 

Ex.  1.  Evaluate  ^"^P^^  when  ^  =  0. 

Pat  /(^)  =  l-cosd,        <!>($)  =  e^. 

Then  /'($)  =  sin  0,  <f>'(e)  =  2  0, 

and  /(O)  =  0,  <^'(0)  =  0. 


49.]  INBETERMINATE  FORMS  83 

Again,  f>(e)=oosO,  <^"(^)=2, 

/"(0)=1,  <^"(0)=2, 

hence  /!"„  kl_^  =  1. 

Ex.  2.   Find     lim   f!±iIl±2^2^^zJ. 
X  -  u  3,4 

lim   e"  +  e-''  +  2  cos  a:  —  4  _    lim   e='  —  e-»  —  2  sin  a? 
a;=:0  ^4  -  x  =  0  4 ^8 

_    Mm   6»  +  e-^  —  2cosa: 
-  X  =  0  22  x2 

_    lim   e»  _  e-x  _^  2  sin  a; 
-x  =  0  24x 

_   lim  e»  +  e-*+  2  cos  a? 
""  a;  i  0  24 


Ex.  3.   Find    lip   ^^-sinxcosar, 
a;  =  u  ™8 


Ex.  4.   Find    "P.  a;^-2xa-4^^+9a:-4. 


+ 

In  this  example,  show  that  x—1  is  a  factor  of  both  numerator  and 
denominator. 

1?^  R     T?;^^    li°i    Stana:- Sar-aH' 
^    Ex.  5.   Find  ^  ^  0 -^ 

In  applying  this  process  to  particular  problems,  the  work 
can  often  be  shortened  by  evaluating  a  non-vanishing  factor 
in  either  numerator  or  denominator  before  performing  the 
differentiation. 

Ex.6.   Find    "ro^^-^>'^^^^. 

The  given  expression  may  be  written 

^i°^    rr_4^2tana:_    lim    ,        ..3   ^i^   tanar 
a:  =  0  ^^     *>'   —^  -x  =  0C^-*;  a;  =  0  "^ 

=  16  . 1  =  16. 


84  DIFFERENTIAL   CALCULUS  [Ch.  V. 

In  general,  if  fCx')=  'f(x)x(p^^^  ^^^  ^^  '^(^)=  ^»  xC^)"?^^' 
<^(a)  =  0,  then 

For 


lim    ^(a^)x(^)  _   lim    ^.  .  .    lim    ^(^)  _  ^,^^.  .  ^'(<^) 


Ex.7.   Find 


lim    sin  x  cos^  a; 


Ex.8.    Find,^!P^/"-^)^^^^(^-^). 
*  —  ^         sin  (x  —  1) 


^  p-i      /  EXERCISES 

^     /  Evaluate  the  following  expressions : 

9.   Lz.£2i£  when  a:  =  0.  15.  ^  "r  e-^ -2  ^^^^^  ^^^ 

sin  X  x^ 

when  a;  =  0.  ,  ^    tan  a:-  sin  x  cos  a: 


sma: 


16.  "^"•^-^^"•^^"'^-^  when  a:  =  0. 


11.  £i=li  when  a:  =  1. 

X  —\  __     sin~^a: 


17.  -ii^i— ±  when  ar  =  0. 

12.  ?l::ilwhen  a:  =  0.  *^°"'^ 

6-  -1 

, «    sin  aa:      ,  ««  a  18.  ^'  ^^°  ^  ~  ^  ~  ^^   when  a:  =  0. 

s"i^ll  a:2  +  X  log  (1  -  X) 

14.   (l±£)i:ii  when  X  =  0.  19-  ^"^l^^"^  when  a:  =  0. 

X  x* 

There  are  other  indeterminate  forms  than  -•      They  are 
g,oo-oo,0«,  r,  ao». 

00 

50.  Evaluation  of  the  indeterminate  form  ^. 

oo 

Let  the  function  ^rr^  become  —  when  x^a.     It  is  re- 
quired  to  find  Jf'^lg. 


49-50.]  INDETERMINATE  FORMS  85 

This  function  can  be  written 


<l>(x)         1    ' 


which  takes  the  form  -  when  x  =  a^  and  can  therefore  be 

evaluated  by  the  preceding  rule. 

When  a;  =  a, 

1  <i,<ix') 


lira   fjx)  ^    lim   i>(j()  _    lira        \.4>(^)J 


f(x) 
If  both  members  be  divided  by   ,  ,  {^  the  equation  becomes 

-.  ^   lim  /(a:)  <^'(a:) 

therefore  l-JZ^l^^^^.  (2) 

This  is  exactly  the  same  result  as  was  obtained  for  the 
form  -;  hence  the  procedure  for  evaluating  the  indetermi- 
nate forms  -,  — ,  is  the  same  in  both  cases. 

When  the  true  value  of  ~r-^  is  0  or  oo,  equation  (1)  is 
satisfied,  independent  of  the  value  of  T77-T;   but  (2)  still 

I'       iCxi 

gives  the  correct  value.     For,  suppose  ^i"    ,)  (  =  0.     Con- 

X  —  a  ^  ^^^ 


sider  the  function 


86  DIFFERENTIAL   CALCULUS  [Cb.  V. 

which  has  the  form  —  when  x  =  a»  and  has  the  determinate 

00 

value  <?,  which  is  not  zero.     Hence  by  (2) 

lim  /(^)  +  gj)(^)  _  /(«)  +  c<^^(«)  _  /^(g)  ■  ^ 

Therefore,  by  subtracting  c, 

lim/C^^ZW 

If  r '^«x^  =  <»i  ^en  jiP  ^  =  0,  which  can  be  treated 
as  the  previous  case. 

51.  Evaluation  of  the  form  oo  •  0. 

Let   the   function   be  (\>{x)  "^(x)^  such  that  <^(a)  =  oo, 
Vr(a)  =  0. 

This  may  be  written  ^^  ^,  which  takes  the  form  -  when 

a  is  substituted  for  a?,  and  therefore  comes  under  the  above 
rule.     (Art.  49.) 

52.  Evaluation  of  the  form  oo  —  oo. 

The  form  oo  —  oo  may  be  finite,  zero,  or  infinite. 
For  instance,  consider  y/a?  +  ax—  x  for  the  value  x  =  cc. 
It  is  of  the  form  oo  —  oo,  but  by  multiplying  and  dividing  by 

^x^  4-  ax  -f  a;  it  becomes  ^ ,  which  has  the  form 

00      1  ^x^  -{-  ax  -\-  X 

5o  when  a;  =  oo. 

Again,  by  dividing  both  terms  by  x,  it  takes  the  form 
.,  which  becomes  ?  when  rr  =  oo. 

4U 


Jl+'  +  l 
>  X 

There   is   here   no   general  rule  of   procedure  as  in  the 
previous  cases,  but  by  means  of  transformations  and  proper 


50-52.]  INDETERMINATE  FORMS  87 

grouping  of  terms  it  is  often  possible  to  bring  it  into  one 
of  the  forms  -,  — .  Frequently  a  function  which  becomes 
oo  —  00  for  a  critical  value  of  x  can  be  put  in  the  form 

u       t 

V         V) 

in  which  v,  w  become  zero.     This  can  be  reduced  to 

uw  —  vt 


which  is  then  of  the  form  -• 


Ex.  1.   Find  3.^5  (sec  x  —  tan  x). 

This  expression  assumes  the  form  00  —  00,  but  can  be  written 

1     _  sin  a-  _  1  —  sin  a: 

cos  X      cos  X  ~     cos  X 

which  is  of  the  form  -,  and  gives  zero  when  evaluated. 
Hence  ^  ^^  (sec  x  —  tan  a:)  =  0. 

Ex.  2.  Prove  ^  ^^n  (sec^a:  —  tan»a:)  =  00,  1,  0,  according  as 


\^ 


EXERCISES  ^ 


Evaluate  the  following  expressions : 

2.  !^S^  when  x  =  0.  6.  ^5£^  when  a;  =  5- 
cot  x  sec  5  a;  2 

3.  —  when  a:  =  00.  7.  (a^  —  x^)  tan  —  when  x  =  a. 

-      tan  X    „i V  0/1       +„^  ^\  o««  o «.  T^kz^r.  «._''' 


when  ar  =  -•  8.    (1  —  tan  ar)  sec  2  a;  when  a:  =  -• 

tan5x  2  4 


88  DIFFERENTIAL   CALCULUS  [Ch.  V. 

y  g    1-loga;  ^YiQn  x  =  0.  '^12.  — ^ i—  when  x  =  l. 

e'  —  e     X  —  1 

yr  10.  ^-^  when  :r  =  00.  N  13-  csc^x  -  1  when  x  =  0. 

J  11.  -i ^  when  a;  =  1.  /!*•  r^  "  r^  ^^^^  ^  =  1* 

^  logx      a:-l  *^  logx      loga; 

53.  Evaluation  of  the  form  1*. 

Let  the  function  u  —  [</>(a;)]'''^'^^  assume  the  form  1*  when 
x  —  a. 

In  order  to  evaluate  this  expression,  take  the  logarithm  of 
both  sides.     Then 

log  u  =  ^(x)  ■  log  ^(x)  =  l2E|M. 

This  expression  assumes  the  form  -  when  a;  =  a,  and  can 

be  evaluated  by  the  method  of  Art.  49. 

If  the  reduced  value  of  this  fraction  be  denoted  by  tw, 
then  log  w  =  m  and  u  =  g"*. 

Note.     The  form  1°  is  not  indeterminate,  but  is  equal  to  1. 
For,  let  [^(x)] '/'(*)  assume  the  form  1°  when  x  —  a. 

Put  u  =  [</>(x)]<^(*). 

Then  logu  =  ^(a:)log[<^(a;)], 

which  equals  zero  when  x  ■=.  a\ 

hence  log  u  =  0,     u  —  e^—\, 

54.  Evaluation  of  the  forms  oo^  0^. 
Let  [^^{xy^'''^  become  oo^  when  x  —  a. 
Put  w  =  [</)(a:)]'^<->. 

Then  log  u  =  y^(x)  log  4>(x)  =  ]2K^^, 


52-54.]  INDETERMINATE  FORMS  89 

This  is  of  the  form  — ,  and  can  be  evaluated  by  the  method 
of  Art.  50.     Similarly  for  the  form  0^. 

Note.     The  form  0*  is  not  indeterminate,  but  is  equal  to  0. 
For  let  u  =  [<^(x)]'l'(^>  become  0"°  when  x  =  a. 

Then         log  u  =  i//(a:)  log  <f}(x)=  —  oo,  and  u  =e-*  =  0. 
Similarly,  the  form  0-*  is  equal  to  co. 

This  completes  the  list  of  ordinary  indeterminate  forms. 
The  evaluation  of  all  of  them  depends  upon  the  same 
principle,  namely,  that  each  form  (or  its  logarithm)  may  be 

brought  to  the  form  -,  and  then  evaluated  by  differentiating 

numerator  and  denominator  separately.  In  finally  letting 
x=  a,  the  two  directions  of  approach  should  be  compared, 
so  as  to  reveal  any  discontinuity  in  the  function. 

EXERCISES 
Evaluate  the  following  indeterminate  forms : 

1.  a:'""'  when  x  =  0. 

2.  (cosaa:)«8«'''=*  when  a:  =  0. 

6 

3.  (1  4-  axy  when  a:  =  0. 

6 

4.  (1  4-  ax)"  when  a:  =  co. 

/1\8in«  /  d  \x 

5.  I  -  1        when  x  =  0.  9,    (  _  4.  1  J    when  x  = 


6. 

^2--^"'-  when  x  =  a. 

7. 

X"  when  x  =  0. 

8. 

1 
o(f  when  x  =  co. 

EXERCISES  ON  CHAPTER  V 
Evaluate  the  following  indeterminate  forms : 

^  1.   ^^^  whena:  =  0.  ^ ^,     ^^^"^     whena:  =  0.    ' 

X  V(e*  -  1)8 

^  2.   -^  when  x  =  0.  ^5.  1^S£  when  x  =  0. 

sin  X  CSC  X 

gmx  _  gma  /  6.     6"*  log  X  whcU  a:  =  00. 

^    3. —  when  a;  =  a.  ^„  y-. . 

(x  —ay  y  1.   X  —  vx^  —  ax  when  a:  =  oo. 


90 


[Ch.  V.  64. 


DIFFERENTIAL   CALCULUS 
8.  (cot  x)""  *  when  a;  =  0. 

10.  1  -  coto;  when  a:  =  0.  ^^     ^t^^^  ^j^^^  ^  ^  ^^ 

11.  -^ 1_  ^hen  a;  =  1.  15.    V2-sinx-cosa:  ^^^^  ^^tt 

x2  -  1      a;  —  1  log  sin  2  X  4 

T  16.  a;  tan  x  —  J  sec  a:  when  a;  =  J* 
2  2 


^  12.  2*  sin  ^  when  a:  =  oo 


13. 


17.    fl2^Ywhena:  =  oo. 

/-        /-        J \    X    I 

^     whena:=6.       is.  x 


X ) ""r 

•  x^  log  [  1  +  -  )  when  a:= oo. 


CHAPTER  VI 
MODE  OF  VARIATION  OF  FUNCTIONS  OF  ONE  VARIABLE 

55.  In  this  chapter  methods  of  exhibiting  the  march  or 
mode  of  variation  of  functions,  as  the  variable  takes  all 
values  in  succession  from  —  oo  to  +  oo,  will  be  discussed. 
Simple  examples  have  been  given  in  Art.  13  of  the  use 
that  can  be  made  of  the  derivative  function  <^'  (a;)  for  this 
purpose. 

The  fundamental  principle  employed  is  that  when  x  in- 
creases through  the  value  a,  ^{x)  increases  through  the 
value  </>(«)  if  </>'(«)  is  positive,  and  that  ^{x)  decreases 
through  the  value  <^(«)  if  <^' («)  is  negative.  Thus  the 
question  of  finding  whether  ^{x)  increases  or  decreases 
through  an  assigned  value  ^(a),  is  reduced  to  determining 
the  sign  of  </>'  (a). 

1.  Find  whether  the  function 

increases  or  decreases  through  the  values  ^(3)  =  2,  ^(0)  =  5,  ^(2)=|, 
^(—  1)=  10,  and  state  at  what  value  of  x  the  function  ceases  to  increase 
and  begins  to  decrease,  or  conversely. 

56.  Turning  values  of  a  function.  It  follows  that  the 
values  of  x  at  which  ^{x)  ceases  to  increase  and  begins  to 
decrease  are  those  at  which  <^'  (x)  changes  sign  from  positive 
to  negative ;  and  that  the  values  of  x  at  which  </>  {x)  ceases 
to  decrease  and  begins  to  increase  are  those  at  which  <^'  (x) 
changes  its  sign  from  negative  to  positive.  In  the  former 
case,  (^  {x)  is  said  to  pass  through  a  Tyvaximum,  in  the  latter, 
a  nbinimum  value. 

91 


92 


DIFFERENTIAL   CALCULUS 


[Ch.  VI. 


Fig.  15. 


Ex.  1.    Find  the  turning  values  of  the  function 
<^  (x)  =  2  a;8  -  3  a;2  -  12  x  +  4, 
and  exhibit  the  mode  of  variation  of  the  function  by  sketch- 
X  ing  the  curve  y  =  <f>{x). 

Here  <f>' (x)=  Qx^  -  Qx  -  12  =  Q(x -^  1)  (x  -  2), 

hence  <f>'  (x)  is  negative  when  x  lies  between  —  1  and  +  2, 
and  positive  for  all  other  values  of  x.  Thus  <fi(x)  increases 
from  X  =  —  x  toa:  =  —  1,  decreases  from  ar  =  — ltoa:  =  2, 
and  increases  from  x  =  2  to  x  =  cc.  Hence  <^(—  1)  is  a 
maximum  value  of  <^  (x),  and  <^  (2)  a  minimum. 

The  general  form  of  the  curve  y  =  <f>(x)  (Fig.  15)  may- 
be inferred  from  the  last  statement,  and  from  the  following  simultaneous 
values  of  x  and  y : 

x  =  -oo,  -2,  -1,0,       1,        2,       3,    4,  00. 
y  =  -  00,      0,     11,  4,  -  9,  -  16,  -  5,  36,  oo. 

Ex.  2.    Exhibit  the  variation  of  the 
function 

<f>(x)  =  ix-l)'-^2, 

especially  its  turning  values. 

Since    <f>'(x)  =  ? , 

3(x-l)^ 

hence  <f>'(x)  changes  sign  at  a:=  1,  being 

negative  when  a:  <  1,  infinite  when  x  =  1, 

and  positive  when  x>l.     Thus  <^(1)  =  2 

is  a   minimum   turning  value  of  <f>(x). 

The  graph  of  the  function  is  as  shown  in  Fig.  16,  with  a  vertical  tangent 

at  the  point  (1,  2). 

Ex.  3.   Examine  for  maxima  and  minima  th6  function 

<t>(x)  =  (x-  1)^  +  1. 

Here      </)'(x)  =  i ? , 

3(x-l)S 

hence  <t>'(x)  never  changes  sign,  but  is 
always  positive.  There  is  accordingly  no 
turning  value.  The  curve  y  =  <f>(x)  has  a 
vertical  tangent  at  the  point  (1,  1),  since 
'-^  =  4*'(x)  is  infinit<*  when  x  =  l.   (Fig.  17.) 


Fig.  16. 


Fio.  17. 


1 


56-58.]  VARIATION  OF  FUNCTIONS  93 

57.  Critical  values  of  the  variable.  It  has  been  shown  that 
the  necessary  and  sufficient  condition  for  a  turning  value  of 
<l>(x)  is  that  <t>'(^x)  shall  change  its  sign.  Now  a  function 
can  change  its  sign  only  when  it  passes  through  zero,  as  in 
Ex.  1  (Art.  56} ,  or  when  its  reciprocal  passes  through  zero, 
as  in  Exs.  2,  3.  In  the  latter  case  it  is  usual  to  say  that  the 
function  passes  through  infinity.  It  is  not  true,  conversely, 
that  a  function  always  changes  its  sign  in  passing  through 
zero  or  infinity,  e.g.,  y  =  x^. 

Nevertheless  all  the  values  of  re,  at  which  (f>\x)  passes 
through  zero  or  infinity,  are  called  critical  values  of  a;,  be- 
cause they  are  to  be  further  examined  to  determine  whether 
<t>'(x)  actually  changes  sign  as  x  passes  through  each  such 
value ;  and  whether,  in  consequence,  4>(x)  passes  through  a 
turning  value. 

For  instance,  in  Ex.  1,  the  derivative  (i>'(x)  vanishes  when 
a;  =  —  1,  and  when  a:  =  2,  and  it  does  not  become  infinite  for 
any  finite  value  of  x.  Thus  the  critical  values  are  —  1,  2, 
both  of  which  give  turning  values  to  (f>(x).  Again,  in 
Exs.  2,  3,  the  critical  value  is  x  =  1,  since  it  makes  <l>'(x') 
infinite ;  it  gives  a  turning  value  to  <t>(^x)  in  Ex.  2,  but  not 
in  Ex.  3. 


58.  Method  of  determining  whether  <{>'(»)  changes  its  sign 

^  »    in  passing  through  zero  or  infinity.     Let  a  be  a  critical  value 

i^   of  a:,  in  other  words  let  (^'(a)  be  either  zero  or  infinite,  and 

let  ^  be  a  very  small  positive  number,  so  that  a  —  h  and  a-\-h 

^re  two  numbers  very  close  to  a,  and  on  opposite  sides  of  it. 

^in  order  to  determine  whether  <^'(a;)  changes  sign  as  x  in- 

Acreases  through  the  value  a,  it  is  only  necessary  to  compare 

^the  signs  of  <^'(a  +  h)  and  <^'(a  —  h).     If  it  is  possible  to 

^  take  h  so  small  that  <^'(a  —  h}  is  positive  and  <^'(a  +  h) 

V    negative,  then  (j)'(ix)  changes  sign  as  x  passes  through  the 


94 


DIFFERENTIAL   CALCULUS 


[Ch.  VI. 


value  «,  and  <t)(x)  passes  through  a  maximum  value  <^(a). 
Similarly,  if  ^'(a  —  A)  is  negative  and  <^'(a  +  h)  positive, 
then  <t>(jc)  passes  through  a  minimum  value  <^(a). 

if  <^'(a  —  h)  and  <^'(a  +  h)  have  the  same  sign,  however 
small  h  may  be,  then  <^(a)  is  not  a  turning  value  of  <^(a;). 

Ex.    Find  the  taming  values  of  the  function 

Here  <l>'(x)=  2{x  -  l)(a:  +  1)8  +  3(x  -  V)\x  +  1)« 

=  (z-l)(a;  +  l)2(5a:-l). 

Hence  4*'(x)  becomes  zero  at  a:  =  —  1,  |,  and  1 ;  it  does  not  becdme 
infinite  for  any  finite  value  of  x. 

Thus,  the  critical  values  are  —  1,  |,  1. 

F 


Fio.  18. 


When  X  =  —  1  —  h^  the  three  factors  of  <^'(^)  ^^^®  ^-l^®  signs  — 
and  when  a:  =  —  1  +  A,  they  become  — 

thus  ^'{x)  does  not  change  sign  as  x  increases  through  —  1 ; 
<^(—  1)=  0  is  not  a  turning  value  of  <^(x). 

When  r  =  ^  —  A,  the  three  factors  of  ^'(•^)  h*^®  signs  - 

and  when  x  =  \  +  h,  they  become  — 

thus  <l>'{x)  changes  sign  from  +  to  —  as  x  increases  through  \y  and 
^(i)  =  1  •  11052  is  a  maximum  value  of  </)(x). 


+    -, 

+  -; 

hence 

+    -, 

+  +; 


68-59.] 


VARIATION  OF  FUNCTIONS 


95 


Finally,  when  x=l  —  h,  the  three  factors  of  <f>'  (x)  have  the  signs  —  +  + , 
and  when  x  =  1  +  h  they  become  +  +  + ; 

thus  <f>'(x)  changes  sign  from  —  to  +  as  a;  increases  through  1,  and 
<^(1)=  0  is  a  minimum  value  of  <f>{x). 

The  deportment  of  the  function  and  its  first  derivative  in  the  vicinity 
of  the  critical  values  may  be  tabulated  as  follows,  in  which  inc.,  dec.  stand 
for  increasing,  decreasing,  respectively : 


1  +h 

+ 
inc. 


The  general  march  of  the  function  may  be  exhibited  graphically  by 
tracing  the  curve  y  =  <f>(x)  (Fig.  18),  using  the  foregoing  result  and 
observing  the  following  simultaneous  values  of  x  and  y : 


X 

-l-Jl 

-1 

-l  +  h 

\-^ 

\ 

l  +  h 

l-h 

1 

<i>'{x) 

+ 

0 

+ 

4- 

0 

- 

- 

0 

<f>(x) 

inc. 

inc. 

iiic. 

max. 

dec. 

dec. 

min. 

y  = 


1,  0,       i,  1,     2,  00. 
0,  1,  1.1...,  0,  27,  00. 


59  Second  method  of  determining  whether  <t>'(i»)  changes 
sign  in  passing  through  zero.  The  following  method  may  be 
employed  when  the  function  and  its  derivatives  are  continu- 
ous in  the  vicinity  of  the  critical  value  x  =  a. 

Suppose,  when  x  increases  through  the  value  a,  that  <f)'(^x) 
changes  sign  from  positive  through  zero  to  negative.  Its 
change  from  positive  to  zero  is  a  decrease,  and  so  is  the 
change  from  zero  to  negative ;  thus  (^'(a:)  is  a  decreasing 
function  at  x  =  a^  and  hence  its  derivative  <t>"(x)  is  nega- 
tive 2it  X—  a. 

On  the  other  hand,  if  <\>\x^  changes  sign  from  negative 
through  zero  to  positive,  it  is  an  increasing  function  and 
(^"(a?)  is  positive  at  a:  =  a;  hence  : 

The  function  <^(a;)  has  a  maximum  value  (f>(a'),  when  <^'(a)  =  0 
and  <j>"(a^  is  negative  ;  <^(a:)  has  a  minimum  value  4>(cl)-,  when 
<^'(a)=  0  and  <^"(a)  is  positive. 


96  DIFFERENTIAL   CALCULUS  [Ch.  VI. 

It  may  happen,  however,  that  <i>"  (ci)  is  also  zero. 

In  this  case,  to  determine  whether  <\>(x)  has  a  turning 
value,  it  is  necessary  to  proceed  to  the  higher  derivatives. 
If  (i>(x)  is  a  maximum,  <t>"(x)  ^^  negative  just  before  vanish- 
ing, and  negative  just  after,  for  the  reason  given  above  ;  but 
the  change  from  negative  to  zero  is  an  increase,  and  the 
•change  from  zero  to  negative  is  a  decrease;  thus  (t>"(x) 
I  changes  from  increasing  to  decreasing  as  x  passes  through  a. 
Hence  <f>"'(x')  changes  sign  from  positive  through  zero  to 
negative,  and  it  follows,  as  before,  that  its  derivative  <^'^(a;) 
is  negative. 

Thus  <^(a)  is  a  maximum  value  of  <t>(x)  if  <^'(a)=0, 
<^'^(a)=0,  </>'"(«)=  0,  <^'''(a)  negative.  Similarly,  <^(a)  is 
a  minimum  value  of  <i>Qc)  if  </>'(a)  =  0,  <^''(a)  =  0,  <^'^'(a)  =  0, 
and  </>'^(a)  positive. 

If  it  happen  that  (^'^(a)  =  0,  it  is  necessary  to  proceed  to 
still  higher  derivatives  to  test  for  turning  values.  The 
result  may  then  be  generalized  as  follows: 

The  function  <\>(x)  has  a  maximum  {or  minimum')  value  at 
x=  a  if  one  or  more  of  the  derivatives  <^'(a),  <f>"{a},  (i>"'{cL) 
vanish  and  if  the  first  one  that  does  not  vanish  is  of  even  order ^ 
and  negative  {or  positive), 

Ex.     Find  the  critical  values  in  the  example  of  Art.  58  by  the  second 
method. 
<^"(:r)  =  (x+l)2(5a:-l)  +  2(x-l)(x+l)(5x-l)+5(a:-l)(z+l)a, 

=  4(5x»  +  3a:a-3x-  1), 
<^"(1)=  16,  hence  ^(1)  is  a  minimum  value  of  <^(a:), 
^"(—  1)  =  0,  hence  it  is  necessary  to  find  <!>'"(—  1)  ; 
<^"'(x)=12(6x«  +  2x-l), 

^'"(— 1)=24,  hence  <^(- 1)  is  neither  a  maximum  nor  a  minimum 
value  of  <f>(j:)- 

Again,  <^"(i)  =  H^  -  l)(i  +  1)*  ^  negative,  hence  <f>(^)  is  a  maximum 
value  of  <ft(x). 


69-60.]  VARIATION   OF  FUNCTIONS  97 

60.  Conditions  for  maxima  and  minima  derived  from  Tay- 
lor's theorem.  In  this  article,  as  in  the  preceding,  the  func- 
tion and  its  derivatives  are  supposed  to  be  continuous  in  the 
vicinity  of  x  =  a;  otherwise  the  method  of  Art.  58  must  be 
used. 

If  <l>Qa)  be  a  maximum  value  of  <^(a;),  it  follows  from  the 
definition  that  </>(«)  is  greater  than  either  of  the  neighboring 
values,  </>(«  +  A),  or  <j>(^a  —  A),  when  h  is  taken  small  enough. 
Hence  <^(a  +  A)—  </>(«)  and  <^(a  —  A)—  </>(«)  are  both 
negative. 

Similarly,  these  expressions  are  both  positive  if  <^(«)  is  a 
minimum  value  of  (f>(^x^. 

Let  <l>(x  +  K)  and  (f>Qc  —  K)  be  expanded  in  powers  of  h  by 
Taylor's  theorem. 

Then  <^(:i:+A)  =  </>(2:)+<^'(a:)A+^^A2  +  *^^A3+..., 

A  .  o  I 

If  X  be  replaced  by  a,  and  ^(a)  transposed,  the  result  is 

The  increment  h  can  now  be  taken  so  small  that  h(i>'(a') 
will  be  numerically  larger  than  the  sum  of  the  remaining 
terms  in  the  second  member  of  either  of  the  last  two  equa- 
tions, and  its  sign  will  therefore  determine  the  sign  of  the 
entire  member.  Since  these  signs  are  opposite  in  the  two 
equations,  <^(a  +  A)  —  <^(a)  and  <^(a  —  A)  —  <^(a)  cannot  have 
the  same  sign  unless  <^'(a)  is  zero,  hence  the  first  condition 
for  a  turning  value  is  <^' (a)  =  0. 


98  DIFFERENTIAL   CALCULUS  [Ch.VI. 

In  case  <f)'(a)=  0  the  preceding  equations  become 

and  h  can  be  taken  so  small  that  the  first  term  on  the  right 
is  numerically  larger  than  either  of  the  second  terms,  hence 
(j>(a  +  K)  —  <^(«)  and  (f>(a  —  K)—  (f>(cL)  are  both  negative  when 
</)"(«)  is  negative,  and  both  positive  when  <j>"(a}  is  positive. 

Thus  <^(«)  is  a  maximum  (or  minimum)  value'  of  <t>{x) 
when  <^'(a)  is  zero  and  <i>" (a)  is  negative  (or  positive). 

If  it  should  happen  that  <i>"  (a)  is  also  zero,  then 

and  by  the  same  reasoning  as  before,  it  follows  that  for  a 
maximum  (or  minimum)  there  are  the  further  conditions 
that  <f>"'(cL)  equals  zero,  and  that  <^*^(«)  is  negative  (or 
positive) . 

Proceeding  in  this  way,  the  general  conclusion  stated  in 
the  last  article  is  evident. 

Ex.  1.  Which  of  the  preceding  examples  can  be  solved  by  the  general 
rule  here  referred  to  ? 

Ex.  2.  Why  was  the  restriction  imposed  upon  <i>'{x)  that  it  should 
change  sign  by  passing  through  zero,  rather  than  by  passing  through 
infinity? 

61.  The  maxima  and  minima  of  any  continuous  function 
occur  alternately.     It  has  been  seen  that  the  maximum  and 


60-62.]  VABIATION  OF  FUNCTIONS  99 

minimum  values  of  a  rational  polynomial  occur  alternately 
when  the  variable  is  continually  increased,  or  diminished. 

This  principle  is  also  true  in  the  case  of  every  continuous 
function  of  a  single  variable.  For,  let  c/>(<x),  4>(h^  be  two 
maximum  values  of  <^(a;),  in  which  a  is  supposed  less  than 
h.  Then,  when  x  =  a-\-  h,  the  function  is  decreasing ;  when 
x=h  —  h,  the  function  is  increasing,  h  being  taken  suffi- 
ciently small  and  positive.  But  in  passing  from  a  decreas- 
ing to  an  increasing  state,  a  continuous  function  must,  at 
some  intermediate  value  of  x,  change  from  decreasing  to 
increasing,,  that  is,  must  pass  through  a  minimum.  Hence, 
between  two  maxima  there  must  be  at  least  one  minimum. 

It  can  be  similarly  proved  that  between  two  minima  there 
must  be  at  least  one  maximum. 

62.  Simplifications  that  do  not  alter  critical  values.  The 
work  of  finding  the  critical  values  of  the  variable,  in  the 
case  of  any  given  function,  may  often  be  simplified  by  means 
of  the  following  self-evident  principles. 

1.  When  c  is  independent  of  x,  any  value  of  x  that  gives 
a  turning  value  to  c(\>(x)  gives  also  a  turning  value  to 
(t>(x);  and  conversely.  These  two  turning  values  are  of 
the  same  or  opposite  kind  according  as  c  is  positive  or 
negative. 

2.  Any  value  of  x  that  gives  a  turning  value  to  c-f-  <l)(x} 
gives  also  a  turning  value  of  the  same  kind  to  (/>(a;);  and 
conversely. 

3.  When  n  is  independent  of  x^  any  value  of  x  that  gives 
a  turning  value  to  [</>(2;)]"  gives  also  a  turning  value  to 
(/>(a;);  and  conversely.  Whether  these  turning  values  are 
of  the  same  or  opposite  kind  depends  on  the  sign  of  n^  and 
also  on  the  sign  of  [^(a^)]""^- 


y\A 


iajL^ 


100  DIFFERENTIAL  CALCULUS  [Ch.  VI. 


EXERCISES 

Find  the  critical  values  of  x  in  the  following  functions,  determine  the 
nature  of  the  function  at  each,  and  obtain  the  graph  of  the  function. 

/ 


1.    M  =  ar(a;2-1). 

6. 

M  =  ar(ar+1)2- 

^     2.    M  =  2  a:8  -  15  a;2  +  36  a:  - 

-4. 

1      7. 

M  =  5  +  12  a:  -  J 

3.   M=  (x-  1)8  (a; -2)2. 

8. 

u  =  !2££. 

^  4.   w  —  sin  X  +  cos  x. 

5.   «=(^^^. 
a-2x 

9. 

X 

u  =  sin^  X  cos8  X. 

^' 


10.  Show  that  a  quadratic  integral  function  always  has  one  maxi- 
mum, or  one  minimum,  but  never  both. 

11.  Show  that  a  cubic  integral  function  has  in  general  both  a  maxi- 
mum and  a  minimum  value,  but  may  have  neither. 

12.  Show  that  the  function  (x  —  by  has  neither  a  maximum  nor  a 
minimum  value. 

^ '  63.  Geometric  problems  in  maxima  and  minima.  The 
theory  of  the  turning  values  of  a  function  has  important 
applications  in  solving  problems  concerning  geometric 
maxima  or  minima,  i.e.,  the  determination  of  the  largest 
or  the  smallest  value  a  magnitude  may  have  while  satisfying 
certain  stated  geometric  conditions. 

The  first  step  is  to  express  the  magnitude  in  question 
algebraically.  If  the'  resulting  expression  contains  more 
than  one  variable,  the  stated  conditions  will  furnish  enough 
relations  between  these  variables,  so  that  all  the  others 
may  be  expressed  in  terms  of  one.  The  expression  to 
be  maximized  or  minimized,  being  thus  made  a  func- 
tion of  a  single  variable,  can  be  treated  by  the  preced- 
ing rules. 


62-63.]  VARIATION  OF  FUNCTIONS  101 

Ex.  1.  Find  the  largest  rectangle  whose  perimeter  is  100.  Let  x,  y 
denote  the  dimensions  of  any  of  the  rectangles  whose  perimeter  is  100. 
The  expression  to  be  maximized  is  the  area 

u  =  xy,  (1) 

in  which  the  variables  x,  y  are  subject  to  the  stated  condition 

2a;+2?/=100, 

le,,  y  =  ^0-x;  (2) 

hence  the  function  to  be  maximized,  expressed  in  terms  of  the  single 
variable  x,  is 

M  =  <^  (a;)  =  a;  (50  -  x)  =  50  a;  -  x"^.  (3) 

The  critical  value  of  x  is  found  from  the  equation 

<^'(a:)  =  50-2ar=0 

to  be  a;  =  25.  When  x  increases  through  this  value,  <^'(x)  changes  sign 
from  positive  to  negative,  and  hence  ^  {x)  is  a  maximum  when  x  =  25. 
Equation  (2)  shows  that  the  corresponding  value  of  y  is  25.  Hence  the 
maximum  rectangle  whose  perimeter  is  100  is  the  square  whose  side  is  25. 

Ex.  2.  If,  from  a  square  piece  of  tin  whose  side  is  a,  a  square  be  cut 
out  at  each  corner,  find  the  side  of  the  latter  square  in  order  that  the 
remainder  may  form  a  box  of  maximum  capacity,  with  open  top. 

Let  a:  be  a  side  of  each  square  cut  out.  Then 
the  bottom  of  the  box  will  be  a  square  whose  side 
is  a  -  2  a:,  and  the  depth  of  the  box  will  be  x. 
Hence  the  volume  is 

v  =  x{a-2xy, 

which  is  to  be  made  a  maximum  by  varying  «. 

Here  ^=  (a  -  2ar)2  -  4a:(a  -  2a:) 

d^  Fio.  19. 

=  (a-2a:)(a-6a:). 

This  derivative  vanishes  when  x  =  -,  and  when  x  =  -.  It  will  be  found, 

2  0 

by  applying  the  usual  test,  that  a:  =  ^  gives  v  the  minimum  value  zero,  and 

O     3 

that  x  =  -  gives  it  the  maximum  value  — ^.      Hence  the  side  of  the 

6  27 

square  to  be  cut  out  is  one  sixth  the  side  of  the  given  square. 


102 


DIFFERENTIAL   CALCULUS 


[Ch.  VI. 


Ex.  3.    Find  the  area  of  the  greatest  rectangle  that  can  be  inscribed 

in  a  given  ellipse. 

An  inscribed  rectangle 
will  evidently  be  sym- 
metric with  regard  to 
the  principal  axes  of  the 
ellipse. 

Let  a,  b  denote  the 
lengths  of  the  semi-axes 
OA,  05  (Fig. 20);  let2.T, 
2y  he  the  dimensions  of 
an  inscribed  rectangle. 
Then   the   area  is 


Fig.  20. 


u  =  4:xy, 


(1) 


in  which  the  variables  x,  y  may  be  regarded  as  the  coordinates  of  the 
vertex  P,  and  are  therefore  subject  to  the  equation  of  the  ellipse 


t+t  =  l 


ft2 


(2) 


It  is  geometrically  evident  that  there  is  some  position  of  P  for  which 
the  inscribed  rectangle  is  a  maximum. 

The  elimination  of  y  from  (1),  by  means  of  (2),  gives  the  function  of 
X  to  be  maximized, 

(3) 


«=i*a:V^^^ 


By  Art.  62,  the  critical  values  of  x  are  not  altered  if  this  function  be 

4  A 
divided  by  the  constant  — ,  and  then  squared.     Hence,  the  values  of  a? 

a 
which  render  u  a  maximum,  give  also  a  maximum  value  to  the  function 


Here 


<f>  {x)  =  x\a^  -  x2)  =  a2x2  -  x^, 
<f>f(x)  =2a^x  -ix»  =  2x(a^-  2x^, 
<l>"(x)  =  2a^-  12x2; 


hence,  by  the  usual  tests,  the  critical  values  x  =  ± — -  render  ^(x),  and 

\/2 
therefore  the  area  u,  a  maximum.     The  corresponding  values  of  y  are 
given  by  (2),  and  the  vertex  P  may  be  at  any  of  the  four  points 
denoted  by 

V2  y/f 


63.] 


VABIATION  OF  FUNCTIONS 


103 


giving  in  each  case  the  same  maximum  inscribed  rectangle,  whose 
dimensions  are  aV2,  by/2,  and  whose  area  is  2  aft,  or  half  that  of  the 
circumscribed  rectangle. 

Ex.  4.    Find  the  greatest  cylinder  that  can  be  cut  from  a  given  right 
cone,  whose  height  is  h,  and  the  radius  of  whose  base  is  a. 

Let  the  cone  be  generated  by  the 
revolution     of     the     triangle     GAB  ^^B 

(Fig.  21),  and  the  inscribed  cylinder 
be  generated  by  the  revolution  of  the 
rectangle  AP. 

Let  OA  =hj  AB  =  ttj  and  let  the 
coordinates  of  P  be  (x,  y).  Then  the 
function  to  be  maximized  is  Try^Qi  —  x) 

subject  to  the  relation  -  —  t'  -       ^^'  ^' 


This  expression  becomes 


V  = 


h^ 


x\h  -  x). 


The  critical  value  of  a;  is  f  A,  and  F  = 


27     ' 


EXERCISES  ON  CHAPTER  VI 

1.  Through  a  given  point  within  an  angle  draw  a  straight  line  which 
shall  cut  off  a  minimum  triangle.  Solve  this  problem  by  the  method  of 
the  calculus,  and  also  by  geometry. 

[Take  given  lines  as  coordinate  axes.] 

2.  The  volume  of  a  cylinder  being  constant,  find  its  form  when  the 
entire  surface  is  a  minimum. 

3.  A  rectangular  court  is  to  be  built  so  as  to  contain  a  given  area  c^, 
and  a  wall  already  constructed  is  available  for  one  of  its  sides.  Find  its 
dimensions  so  that  the  expense  incurred  may  be  the  least  possible. 

4.  The  sum  of  the  surfaces  of  a  sphere  and  a  cube  is  given.  How  do 
their  volumes  compare  when  the  sum  of  their  volumes  is  a  minimum  ? 

5.  What  is  the  length  of  the  axis  of  the  maximum  parabola  which 
can  be  cut  from  a  given  right  circular  cone,  given  that  the  area  of  a 
parabola  is  equal  to  two  thirds  of  the  product  of  its  base  and  altitude  ? 

6.  Determine  the  greatest  rectangle  which  can  be  inscribed  in  a  given, 
triangle  whose  base  is  2  6  and  whose  altitude  is  a. 


.ir . 


104  DIFFERENTIAL   CALCULUS  [Ch.  VI.  63. 

7.  The  flame  of  a  candle  is  directly  over  the  center  of  a  circle  whose 
radius  is  5  inches.  What  ought  to  be  the  height  of  the  flame  above  the 
plane  of  the  circle  so  as  to  illuminate  the  circumference  as  much  as  pos- 
sible, supposing  the  intensity  of  the  light  to  vary  directly  as  the  sine  of 
the  angle  under  which  it  strikes  the  illuminated  surface,  and  inversely 
as  the  square  of  its  distance  from  the  illuminated  point  ? 

y^  8.  A  rectangular  piece  of  pasteboard  30  inches  long  and  14  inches 
wide  has  a  square  cut  out  at  each  corner.  Find  the  side  of  this  square 
so  that  the  remainder  may  form  a  box  of  maximum  contents. 

9.  Find  the  altitude  of  the  right  cylinder  of  greatest  volume  in- 
scribed in  a  sphere  whose  radius  is  r. 

10.  Through  the  point  (a,  b)  a  line  is  drawn  such  that  the  part  inter- 
cepted between  the  rectangular  coordinate  axes  is  a  minimum.  Find  its 
length. 

^  11.  Given  the  slant  height  a  of  a  right  cone ;  find  its  altitude  when 
the  volume  is  a  maximum. 

\  12.  The  radius  of  a  circular  piece  of  paper  is  r.  Find  the  arc  of  the 
sector  which  must  be  cut  from  it  so  that  the  remaining  sector  may  form 
the  convex  surface  of  a  cone  of  maximum  volume. 

13.  Find  relation  between  length  of  circular  arc  and  radius  in  order 
that  the  area  of  a  circular  sector  of  given  perimeter  should  be  a  maximum . 

14.  On  the  line  joining  the  centers  of  two  spheres  of  radii  r,  R,  find 
the  distance  of  the  point  from  the  center  of  the  first  sphere  from  which 
the  maximum  of  spherical  surface  is  visible. 

15.  Describe  a  circle  with  its  center  on  a  given  circle  so  that  the 
length  of  the  arc  intercepted  within  the  given  circle  shall  be  a  maximum. 


CHAPTER  VII 
RATES  AND  DIFFERENTIALS 

64.  Rates.  Time  as  independent  variable.  Suppose  a  par- 
ticle P  is  moving  in  any  path,  straight  or  curved,  and  let  s 
be  the  number  of  space  units  passed  over  in  t  seconds.  Then 
s  may  be  taken  as  the  dependent  variable,  and  t  as  the  inde- 
pendent variable. 

If  As  be  the  number  of  space  units  described  in  the  addi- 
tional time  A^  seconds,  then  the  average  velocity  of  P  during 

As 
the  time  A^  is  — ;  that  is,  the  average  number  of  space  units 

described  per  second  during  the  interval. 

The  velocity  of  P  is  said  to  be  uniform  if  its  average 

As 
velocity  —  is  the  same  for  all  intervals  A^.      The   actual 

velocity  of  P  at  any  instant  of  time  t  is  the  limit  which  the 
average  velocity  approaches  as  A^  is  made  to  approach  zero 
as  a  limit. 

Thus  t;=    1/^^^=^ 

^t  =  ^M      dt 

is  the  actual  velocity  of  P  at  the  time  denoted  by  t.  It  is 
evidently  the  number  of  space  units  that  would  be  passed 
over  in  the  next  second  if  the  velocity  remained  uniform 
from  the  time  t  to  the  time  ^  -}- 1. 

It  may  be  observed  that  if  the  more  general  term,  "  rate 
of  change,"  be  substituted  for  the  word  "velocity,"  the 
above  statements  will  apply  to  any  quantity  that  varies  with 
the  time,  whether  it  be  length,  volume,  strength  of  current, 

105 


106  DIFFERENTIAL   CALCULUS  [Ch.  VII. 

or  any  other  function  of  the  time.      For  instance,  let  the 
quantity  of  an  electric  current  be  C  at  the  time  ^,  and  C-{-AO 

at  the  time  t  +  A^.     Then  the  average  rate  of  change  of  cur- 

AC 
rent  in  the  interval  A^  is ;   this  is  the  average  increase 

in  current-units  per  second.     And  the  actual  rate  of  change  at 
the  instant  denoted  by  t  is 

lim    AO^dO 
At  =  0 /^t       dt' 

This  is  the  number  of  current-units  that  would  be  gained 
in  the  next  second  if  the  rate  of  gain  were  uniform. from  the 
time  t  to  the  time  t-\-l.     Since,  by  Art.  14, 

dy  _dy  ^  dx 
dx      dt     dt 

hence  -^  measures  the  ratio  of  the  rates  of  change  of  y  and 

ax 
of  X, 

It  follows  that  the  result  of  differentiating 

y=f(?d  (1) 

may  be  written  in  either  of  the  forms 

!=/'(-),  (2) 

The  latter  form  is  often  convenient,  and  may  also  be 
obtained  directly  from  (1)  by  differentiating  both  sides 
with  regard  to  t.  It  may  be  read :  the  rate  of  change  of 
y  is  f'(x)  times  the  rate  of  change  of  x. 

Returning  to  the  illustration  of  a  moving  point  P,  let  its 

coordinates  at  time  t  hQ  x  and  y.     Then  —  measures  the 
rate  of  change  of  the  a;-coordinate. 

Since  velocity  has  been  defined  as  the  rate  at  which  a  point 


64.] 


RATES  AND  DIFFERENTIALS 


107 


is  moving,  the  rate  —  may  be  called  the  velocity  which  the 

Cit 

point  P  has  in  the  direction  of  the  a;-axis,  or,  more  briefly, 
the  rc-component  of  the  velocity  of  P. 

It  was  shown  on  p.  105  that  the  actual  velocity  at  any 
instant  t  is  equal  to  the  space  that  would  be  passed  over  in 
a  unit  of  time,  provided  the  velocity  were  uniform  during 

that  unit.     Accordingly,  the   a;-component   of   velocity   — - 

at 

may  be  represented  by  the  distance  FA  (Fig.  22)  which  P 
would  pass  over  in  the  direction  of  the  a;-axis  during  a  unit 
of  time  if  the  velocity  remained  uniform. 

Similarly  -^  is  the  y-component  of  the  velocity  of  P,  and 

may  be  represented  by  the  distance  PB. 

ds 
The  velocity  —  of  P  along  the  curve  can  be  represented 

civ 

by  the  distance  P(7,  measured  on  the  tangent  line  to  the 
curve  at  P.     It  is  evident  that 
PC  is  the  diagonal  of  the  rec- 
tangle PA,  PB. 

Since  PC^  =  PA^  +  P&, 

it  follows  that 

m-m-m-  <•> 


rdS. 

dt 


dx 


dt 


Fig.  22. 


Ex.  1.   If  a  point  describe  the  straight  line  3  x  +  4  ?/  =  5,  and  if  x 
increase  h  units  per  second,  find  the  rates  of  increase  of  y  and  of  s. 

2^  =  1 -far, 

dy  _     S  dx 
dt  ~     4:dt' 

dx 
dt 

it  follows  that  ^  =  -ih,      ^ 
dt  ^  dt 


Since 
hence 

When 


108  DIFFERENTIAL   CALCULUS  [Ch.  VII. 

Ex.  2.  A  point  describes  the  parabola  y"^  =  12  x  in  such  a  way  that 
when  a;  =  3  the  abscissa  is  increasing  at  the  rate  of  2  feet  per  second ;  at 
what  rate  is  y  then  increasing?    Find  also  the  rate  of  increase  of  s. 


Since 

y^=12x, 

then 

2y^=12^, 
dt           dt 

dy_^dx_       6      dx, 
dt      y  dt       Vl2  X  dt  ' 

hence,  when  x  = 

=  3, 

and 

—  =2,  it  follows  that  ^y 
dt                                    dt 

±2. 

^^^'-    {%'-  {llT^  (I)  '  ^--  I  =  ^^  '-'  rer  second. 

Ex.  3.  A  person  is  walking  toward  the  foot  of  a  tower  on  a  horizontal 
plane  at  the  rate  of  5  miles  per  hour ;  at  what  rate  is  he  approaching  the 
top,  which  is  60  feet  high,  when  he  is  80  feet  from  the  bottom? 

Let  X  be  the  distance  from  the  foot  of  the  tower  at  time  t,  and  y  the. 
distance  from  the  top  at  the  same  time.     Then 
x^  +  60-2  =  y\ 

and  x^  =  y^- 

dt      ^  dt 

When  a:  is  80  feet,  y  is  100  feet ;  hence  if  —  is  5  miles  per  hour,  -^ 
•    .      ..  .  ^^  dt  ^  dt 

IS  4  miles  per  hour. 

65.  Abbreviated  notation  for  rates.  When,  as  in  the  above 
examples,  a  time  derivative  is  a  factor  of  each  member  of  an 
equation,  it  is  usually  convenient  to  write,  instead  of  the 

symbols  --^,  -^,  the  abbreviations  dx  and  dy,  for  the  rates 
dt    dt 

of  change  of  the  variables  x  and  y.     Thus  the  result  of 

differentiating  v     ^^  n  ,  /1^ 

/=/(a^)  (1) 

may  be  written  in  either  of  the  forms 


dy  __ 
dx 


f'(x),  (2) 


|=/'(.)|,  (3) 

dy=f'ix-)dx.  (4) 


64-65.]  BATES  AND  DIFFERENTIALS  109 

It  is  to  be  observed  that  the  last  form  is  not  to  be  re- 
garded as  derived  from  equation  (2)  by  separation  of  the 

symbols  dv,  dx;   for  the  derivative  -^  has  been  defined  as 

dx 

the  result  of  performing  upon  «/  an  indicated  operation  rep- 
resented by  the  symbol  — ;  and  thus  the  di/  and  dx  of  the 

7  (XX 

symbol  -^  have  been  given  no  separate  meaning.     The  di/ 
dx 

and  dx  of  equation  (4)  stand  for  the  rates,  or  time  deriva- 
tives, -^  and  —  occurring  in  (3),  while  the  latter  equation 
at  ctt 

is  itself  obtained  from  (1)  by  differentiation  with  regard 
to  t,  by  Art.  14. 

In  case  the  dependence  of  ^  upon  x  be  not  indicated  by  a 
functional  operation  /,  equations  (3),  (4)  take  the  form 

dy  _dy  dx 
dt       dx  dt 

dy  =  -^  dx. 
dx 

In   the   abbreviated   notation,    equation   (4)    of   the    last 

article  is  written 

ds^  =  dx^  +  dy'^. 

Ex.  1.  A  point  describing  the  parabola  y^  —  lpx  is  moving  at  the 
time  t  with  a  velocity  of  v  feet  per  second.  Find  the  rate  of  increase 
of  the  coordinates  x  and  y  at  the  same  instant. 

Differentiating  the  given  equation  with  regard  to  f, 

ydy  =  pdx. 
But  dx,  dy  also  satisfy  the  relation 

rfa;2  +  dy'^  =  u^ ; 
hence,  by  solving  these  simultaneous  equations, 

dx  =  —  ^        V,     dy  — ^ y,  in  feet  per  second. 


110  DIFFERENTIAL   CALCULUS  [Ch.  VII. 

Ex.  2.  A  vertical  wheel  of  radius  10  ft.  is  making  5  revolutions  per 
second  about  a  fixed  axis.  Find  the  horizontal  and  vertical  velocities  of 
a  point  oil  the  circumference  situated  30°  from  the  horizontaL 

Since  a:  =  10  cos  6,    y  =  lO  sin  ^, 

then  dx  =  -10  sin  Odd,    dy  =  10  cos  OdO. 

But  f?^  =  10  TT  =  31.416  radians  per  second, 

hence  dx  =  —  314.16  sin  ^  =  —  157.08  feet  per  second, 

and  dy  =  314.16  cos  $  =  272.06  feet  per  second. 

Ex.  3.  Trace  the  changes  in  the  horizontal  and  vertical  velocity  in  a 
complete  revolution. 

66.  Differentials  often  substituted  for  rates.  The  symbols 
dx,  dy  have  been  defined  above  as  the  rates  of  change  of  x 
and  y  per  second. 

Sometimes,  however,  they  may  conveniently  be  allowed 
to  stand  for  any  two  numbers,  large  or  small,  that  are  pro- 
portional to  these  rates;  the  equations,  being  homogeneous 
in  them,  will  not  be  affected.  It  is  usual  in  such  cases  to 
speak  of  the  numbers  dx  and  dy  by  the  more  general  name 
of  differentials;  they  may  then  be  either  the  rates  them- 
selves, or  any  two  numbers  in  the  same  ratio. 

This  will  be  especially  convenient  in  problems  in  which 
the  time  variable  is  not  explicitly  mentioned. 

Ex.  1.  When  x  increases  from  45°  to  45°  15',  find  the  increase  of 
logjo  sin  x,  assuming  that  the  ratio  of  the  rates  of  change  of  the  function 
and  the  variable  remains  sensibly  constant  throughout  the  short  interval. 

Here  dy  =  logjo^  .  cot  xdx  =  .4343  cot  xdx  =  .4343  dx. 

Let  dx  =  15'  =  .004363  radians. 

Then  dy  =  .001895, 

which  is  the  approximate  increment  of  log,o  sin  x. 

But  log,o  sin  45°  =  -  J  log  2  =  -  .150516, 

therefore  log^o  sin  46°  15'  =  -  .148620. 


65-66.]  BATES  AND  DIFFERENTIALS  111 

Ex.  2.  Expanding  logj^  sin  {x  +  h)  as  far  as  A^  by  Taylor's  theorem, 
and  then  putting  x  =  .785398,  h  =  .004363,  show  what  is  the  error  made 
by  neglecting  the  thii'd  term,  as  was  done  in  Ex.  1. 

Ex.  3.   When  x  varies  from  60°  to  60°  10',  find  the  increase  in  sin  x. 

Ex.  4.  Show  that  log^QX  increases  more  slowly  than  ar,  when  x  >  logj^e, 
that  is,  X  >  .4343. 

Ex.  5.  Two  sides  a,  6  of  a  triangle  are  measured,  and  also  the  in- 
cluded angle  C;  find  the  error  in  the  computed  length  of  the  third  side 
c  due  to  a  small  error  in  the  observed  angle  C. 

[Differentiate  the  equation  c^  =  a^  +  b^  ~2ab  cos  C,  regarding  a,  b  as 
constant-! 
/ 

Ex.  6.    A  vessel  is  sailing  northwest  at  the  rate  of  10  miles  per  hour. 

At  what  rate  is  she  making  north  latitude  ? 

^  Ex.  7.  In  the  parabola  y^  =  12  x,  find  the  point  at  which  the  ordinate 
and  abscissa  are  increasing  equally. 

Ex.  8.  At  what  part  of  the  first  quadrant  does  the  angle  increase 
twice  as  fast  as  its  sine? 

/^  Ex.  9.  Find  the  rate  of  change  in  the  area  of  a  square  when  the  side 
b  is  increasing  at  a  ft.  per  second. 

r  Ex.  10.  In  the  function  y  =  2  x^  -\-  Q,  what  is  the  value  of  x  at  the 
point  where  y  increases  24  times  as  fast  as  x  ? 

l/  Ex.  11.  A  circular  plate  of  metal  expands  by  heat  so  that  its  diameter 
increases  uniformly  at  the  rate  of  2  inches  per  second ;  at  what  rate  is 
the  surface  increasing  when  the  diameter  is  5  inches? 

/  Ex.  12.  What  is  the  value  of  x  at  the  point  at  which  x^  —  5  x^ -{■  17  x 
and  x^  —  S  X  change  at  the  same  rate? 


/ 


Ex.  13.  Find  the  points  at  which  the  rate  of  change  of  the  ordinate 
7/  =  x^  —  6  a:2  +  3  ar  +  5  is  equal  to  the  rate  of  change  of  the  slope  of  the 
tangent  to  the  curve. 

jI  Ex.  14.  The  relation  between  s,  the  space  through  which  a  body  falls, 
and  t,  the  time  of  falling,  is  s-16t^;  show  that  the  velocity  is  equal 
to  32  t. 

The  rate  of  change  of  velocity  is  called  acceleration ;  show  that  the 
acceleration  of  the  falling  body  is  a  constant. 


112  DIFFERENTIAL   CALCULUS  [Ch.  VII.  66. 

.  Ex.  15.  A  body  moves  according  to  the  law  s  =  cos  (nt  +  e).  Show 
that  its  acceleration  is  proportional  to  the  space  through  which  it  has 
moved. 

Ex.  16.  If  a  body  be  projected  upwards  in  a  vacuum  with  an  initial 
velocity  Vq,  to  what  height  will  it  rise,  and  what  will  be  the  time  of 
ascent  ? 

^  Ex.  17.   A  body  is  projected  upwards  with  a  velocity  of  a  feet  per 
second.     After  what  time  will  it  return  ? 

v/  Ex.  18.   If  A  be  the  area  of  a  circle  of  radius  x,  show  that  the  circum- 

dA 

ference  is Interpret  this  fact  geometrically. 

dx 

^  Ex.  19.  A  point  describing  the  circle  x^  +  y^  =  25  passes  through 
(3,  4)  with  a  velocity  of  20  feet  per  second.  Find  its  compon^it  veloci- 
ties parallel  to  the  axes. 


CHAPTER   VIII 

DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES 

Thus  far  only  functions  of  a  single  variable  have  been 
considered.  The  present  chapter  will  be  devoted  to  the 
study  of  functions  of  two  independent  variables  x^  y.  They 
will  be  represented  by  the  symbol 

If  the  simultaneous  values  of  the  three  variables  a:,  y,  z  be 
represented  as  the  rectangular  coordinates  of  a  point  in  space, 
the  locus  of  all  such  points  is  a  surface  having  the  equation 

67.  Definition  of  continuity.  A  function  z  oi  x  and  ^, 
2  =  /(ic,  ?/),  is  said  to  be  continuous  in  the  vicinity  of  any 
point  (a,  5)  when  /(a,  5)  is  real,  finite,  and  determinate,  and 
such  that 

however  A  and  h  approach  zero. 

When  a  pair  of  values  a,  h  exists  at  which  any  one  of  these 
properties  does  not  hold,  the  function  is  said  to  be  discon- 
tinuous at  the  point  (a,  6). 

£.a.,  let  2  =  ^^tl. 

x-y 

When  a:  =  0,  then  z  =  —  1  for  every  value  of  y ;  when  y  =  0  then 
a  =  +  1  for  every  value  of  x.    In  general,  if  y  =  wa:, 

1  —  m 

and  z  may  be  made  to  have  any  value  whatever  at  (0,  0)  by  giving  an 
appropriate  value  to  m. 

113 


114  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 

68.   Partial  differentiation.     If  in  the  function 

a  fixed  value  y-^  be  given  to  ?/,  then 

is  a  function  of  x  only,  and  the  rate  of  change  in  z  caused 
by  a  change  in  x  is  expressed  by 

dz  =  ^dx,  (1) 

dx 

in  which  —  is  obtained  on  the  supposition  that  y  is  constant. 
dx 

To  indicate  this  fact  without  the  qualifying  verbal  state- 
ment, equation  (1)  will  be  written  in  the  form 

d^  =  ^Ux.  (2) 

dx 

The  symbol  —  represents  the  result  obtained  by  differ- 
entiating z  with  regard  to  x^  the  variable  y  being  treated  as 
a  constant;  it  is  called  the  partial  derivative  of  z  with 
regard  to  x. 

From  the  definition  of  differentiation.  Art.  11,  the  partial 
derivative  is  the  result  of  the  indicated  operation 

50^    lim    f(x  +  Aa:,  .y)  —f(x,  y). 
dx      ^^  =  ^  Aa: 

Similarly,  the  symbol  —  represents    the   result   obtained 

by  differentiating  z  with  regard  to  y,  the  variable  x  being 
treated  as  a  constant ;  it  is  called  the  partial  derivative  of  z 
with  regard  to  y. 

The  partial  derivative  of  z  with  regard  to  y  is  accordingly 
the  result  of  the  indicated  operation 


68.]  FUNCTIONS   OF  TWO    VARIABLES  115 

Bz_^     lim    fix,y-\-Ai/)-f(x,^} 

dgZ  =  —  dx  is  called  the  partial  x-differential  of  2,  and 

ox 

dz 
dyZ  =  —-dy\^  called  the  partial  y-differential  of  z. 

if 

Geometrically,  the  two  equations 

define  the  curve  of  section  of  the  surface  z=f(x^y)  made 

by  the  plane  y  —  y^     The  derivative  —  defines  the  slope  of 
the  tangent  line  to  this  curve. 

Similarly,  when  rr  has  a  given  constant  value,  x  =  x^^  the 

partial  derivative  —  is  the  expression  for  the  slope  of  the 
dy 

tangent  to  the  curve  cut  from  the  surface   z^fQc^y')  by 
the  plane  x  =  x-^. 

The  equations  of  these  two  tangent  lines  at  the  point 
(p^v  Vv  ^1)  are 

y  =  y^,  z-z^  =  ^(x-x{), 

dz 
x  =  x^,  z-z^=-A.(y-y{), 

and  hence  the  equation  of  the  plane  containing  these  two 
intersecting  lines  is 

The   plane   is   called   the   tangent  plane   to   the   surface 
^  =f(p^^  y)  at  the  point  {xy,  y^  z^. 


116  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 


EXERCISES 

1.  Given  u  =  x^  +  Z  ^V  -  7  xy\  prove  that  a:^  +  y  ^  =  4 1*. 

dx        dy 

2.  Given  u  =  tan-i  ^,  show  that  a:  ^  +  j/  ^  =  0. 

a;  dx      ^  dy 

3.  M  =  log  (e-  +  eO  ;  find  ^  +  ^. 

^    4.   M  =  sinx3/;   find  ^  +  ^. 
aa;     dy 

'     5.   M  =  log  (x  +  Va:2  +  ?/2)  ;   find  a:  ^  +  y  ^. 

aa:      ay 

6.  u  =  log  (tan  x  +  tan  y  +  tan  z)  ;  show  that 

sin  2a:^  +  sin  2^/^  + sin  23  ^  =  2. 
dx  dy  dz 

7.  M  =  log(a:  +  w);  show  that  ^  +  ^  = -• 
^  ^^       ^^  dx     dy     e- 

69.  Total  differential.  If  both  x  and  y  be  allowed  to  vary 
in  the  function  z  —  f{x^  y),  the  first  question  that  naturally 
arises  is  to  determine  the  meaning  of  the  differential  of  2. 

Let  2i  =/(a^r  Vx)-' 

and  Zj  +  A2  =  /(ajj  +  Aa:,  i/j  +  A!/) 

be  two  values  of  the  function  corresponding  to  the  two  pairs 
of  values  of  the  variables  Xy,  y^  and  x-^  +  Aa:,  y^  +  Az/. 
The  difference 

Az  =/(a:i  4-  Aa;,  y^  +  ^y)-f(x^,  y{) 

may  be  regarded  as  composed  of  two  parts,  the  first  part 
beUig  the  increment  which  z  takes  when  x  changes  from  x^ 
to  ajj  +  Aa:,  while  y  remains  constant  (y  =  yj),  and  the  sec- 
ond part  being  the  additional  increment  which  z  takes  when 


68-69.]  FUNCTIONS  OF  TWO   VARIABLES  117 

t/  changes  from  i/j  to  ^^  +  Ay,  while  x  remains  constant 
(^x  =  x^  +  Ax~).     The  increment  Az  may  then  be  written 

Az  =f(x^  +  Ax,  y^  +  A^^)  -f(x^  +  Ax,  y{) 

+/(^i  +  Aa:,  yi)-fCx^,  yO 

^/(^i  +  Ax,  y^  +  Ay)  -/(a^i  +  Ax,  y^ 

Ay  ^ 

/(^l  +  Ax,  y^-f(^x^,  y^-)  ^^ 
Ax 

From  the  theorem  of  mean  value,  Art.  45,  the  last  equation 
may  be  written 

Az  =  — /(a^i  +  OAx,  y{)Ax  +  —fC^i  +  Aa;,  y^  +  O^Ay^Ay. 
dx  ay 

It  represents  the  actual  increment  Az  which  the  dependent 
variable  z  takes  when  the  independent  variables  x  and  y  take 
the  increments  Ax  and  Ay. 

In  the  preceding  equation  let  Ax,  Ay,  Az  be  replaced  by 
€  •  dx,  €  •  dy,  € '  dz  respectively,  in  which  dx,  dy  are  entirely 
arbitrary.  After  removing  the  common  factor  €,  let  € 
approach  zero.     The  result  is 

d^  =  ^f(^dx  +  ^^^^d2,.  (1) 

The  differential  dz  defined  by  this  equation  is  called  the 
total  differential  of  z.  It  is  not  an  actual  increment  of  z, 
but  the  increment  which  z  would  take  if  the  change  con- 
tinued uniform  while  x  changed  from  x-^^  to  x-^  +  dx  and  y 
changed  from  y^  to  y^  +  dy.  Geometrically  speaking  this  is 
the  increment  which  z  would  take  if  the  point  (x,  y,  2)  should 
move  from  the  position  (x^,  y^  ^i)  in  the  tangent  plane  of 
the  surface  z  =  fix,  y)  instead  of  on  the  surface  itself. 


118  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 

Equation  (1)  may  be  written  in  the  form 

dz  =  —  dx  -\ dy, 

dx  dy    ^ 

from  which  the  following  theorem  can  be  stated :  the  total 

differential  of  a  function  of  two  variables  is  equal  to  the  sum 

of  the  partial  differentials. 

The  same  method  can  be  directly  applied  to  functions  of 

three  or  more  variables.      Thus,  if  2  be  a  function  of  the 

variables  a;,  ^,  w, 

z  =  <l)(x,  y,  u), 

then  dz  =  ^-idx  +  ^dy-^^du. 

dx  dy  du 

Ex.  1.   Given  z  =  axy^  +  bx^y  +  cx^  +  ey, 

then  dz  =  (ay"^  -\-2bxy  +  3  cx^)dx  +  (2  axy  +  bx^  +  e)dy. 

Ex.  2.   Given  z  =  x*,  then  d^  =  yx^-^dx,  and  dyZ  =  x^  log  x  dy. 
Hence  dz  =  yx^~^dx  +  x"  log  x  dy. 

Ex.  3.   Given  u  =  tan-i?^,  show  that  du  =  """^f  ~  ^/^. 
x  x^  -\-  y^ 

Ex.  4.  Assuming  the  characteristic  equation  of  a  perfect  gas,  vp  =  Kt, 
in  which  v  is  volume,  p  pressure,  t  absolute  temperature,  and  R  a  constant, 
express  each  of  the  differentials  dv,  dp,  dt,  in  terms  of  the  other  two. 

Ex.  5.  A  particle  moves  on  the  spherical  surface  x^  -{■  y^  -\-  z^  =  a^  in 
a  vertical  meridian  plane  inclined  at  an  angle  of  60°  to  the  zx  plane. 

If  the  x-component  of  its  velocity  be  ^  a  per  second  when  x  =  \a,  find 
the  y-component  and  the  ^-component  velocity. 


Since 


then  dz  =  — 


xdx ydy 


y/a^  -  x^-  y^      Va^  -  x^  -  y^ 
But  since  dx  =  ^a,  and  the  equation  of  the  given  meridian  plane  is 
y  =  X  tan  60^  hence  dy  =  dxy/S  =  ^  V3,  and  y  =  ^.    Therefore 

(fa  =  --^-^  =  -^  in  feet  per  second. 


69-70.]  FUNCTIONS  OF  TWO   VARIABLES  119 

70.  Language  of  differentials.  The  results  of  the  preced- 
ing articles  may  be  stated  thus  : 

The  partial  ^-differential  due  to  a  change  in  x  is  equal  to 
the  ri:-differential  multiplied  by  the  partial  a;-derivative. 

The  partial  ;3-differential  due  to  a  change  in  ^  is  equal  to 
the  ^/-differential  multiplied  by  the  partial  ^-derivative. 

The  total  ^-differential  is  equal  to  the  sum  of  the  partial 
^-differentials. 

One  advantage  of  writing  the  equation  in  the  differential 
form  is  that  it  may  be  divided  when  necessary  by  the  dif- 
ferential of  any  other  variable  s,  to  which  x  and  y  may  be 
related,  and  then,  remembering  that  the  ratio  of  two  differ- 
entials (or  rates)  may  be  expressed  as  a  derivative,  the 
equation  would  become 

dz_dzdx^  d^^ 
ds      dx  ds      dy  ds 

In  particular,  if  y  be  not  independent,  but  is  a  function  of 
x,  then  s  may  be  chosen  as  x  itself,  and  the  preceding  equa- 
tion becomes  ,        ^ 

dz_oz^.dz     dj£ 

dx     dx     dy    dx 
If  the  functional  relation  between  x  and  y  be  given, 

y  =  ^(^)^ 

dz 
then  the  same  result  would  be   obtained,  whether   -—  be 

dx 

determined  by  the  present  method,  or  y  be  first  eliminated 

from  the  relation 

2'=/(^,  «^), 

and  the  resulting  equation  be  differentiated  as  to  x.     The 
method  of  this  article  frequently  shortens  the  process. 

It  is  here  well  to  note  the  difference  between  —  and  -— . 

dx  dx 

The  former  is  the  partial  derivative  of   the  functional  ex- 


120  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 

pression  for  z  with  regard  to  a:,  on  the  supposition  that  y 
is  constant.  The  latter  is  the  total  derivative  of  z  with 
regard  to  a;,  when  account  is  taken  of  the  fact  that  y  is 
itself  a  function  of  x, 

Ex.  1.   Given  z=  Va;^  +  y\  y  =  \ogx',  find  — • 

dx 


(Iz 


^, 


dx      y/x^  +  V*      y/x^  4-  y^  ^"^ 

rfar     a;' 

hence  ^Il^_^±]L.. 

fi^     xy/x^  +  y^ 

Ex.  2.   If  2  =  tan-i  JL  and  ^x'^-{-y^=l,  show  that  —  =  — • 

71.  Differentiation  of  implicit  functions.  If  in  the  relation 
z—f{x^y)^  z  be  assumed  to  be  constant,  then 

hence  ^dx-\-^dy=0,  (1) 

Bx  By  ^  ^  ^ 

from  which  ^  =  _|£.  (2) 

dx  ^ 

dy 

In  all  such  cases  either  variable  is  an  implicit  function  of 
the  other,  and  thus  the  last  equation  furnishes  a  rule  for 
finding  the  derivative  of  an  implicit  function. 

Ex.  1.   Given  x*  +  y«  +  3  axy  =  c,  find  -p 

Since    (3a:^  +  3a,)  +  (3,«  +  3a.)^^  =  0,   |  =  -^. 

Ex.2.  /(ax+%)=c;  |^=  a/(a:r  +  6y)  ;  ^=  6/'(«a:  +  6y);  ^=-  ^^ 

Ex.  3.    If  aa:2  +  2  ^ary  +  fty"  +  2  ^rar  +  2/y  +  c  =  0,  find  ^ 

Ex.  4.  Given  x*  -  w«  =  c,  find  ^. 

ax 


70-73.]  FUNCTIONS   OF  TWO   VARIABLES  121 

It  is  to  be  noticed  that  the  result  of  differentiating  any 
implicit  function  of  x,  y  by  the  method  of  the  present  article 
agrees  with  the  result  of  differentiation  according  to  the 
rules  of  Chapter  II.     Compare  Ex.  2,  p.  53. 

72.  Successive    partial    differentiation.      The  expressions 

— ,  —  which  were  defined  in  Art.  68  are  functions  of  both 
bx    dy 

X  and  y. 

If  —  be  differentiated  partially  as  to  x^  the  result  is  written 

This  expression  is  called  the  second  partial  derivative  of 
z  as  to  X, 

Similarly,  the  results  of  the  operations  indicated  by 

dy\dx/    dx\dy/    dy\dy) 

S^z  d^z        b^z 

are  written  ,   ,   — r,  respectively. 

by  dx     dx  dy     dy^ 

Beginning  with  the  left,  these  expressions  are  called  the 
second  partial  derivative  of  2  as  to  a;  and  y^  the  second  par- 
tial derivative  of  z  SiS  to  y  and  x,  and  the  second  partial 
derivative  of  z  as  to  y. 

73.  Order  of  differentiation  indifferent. 
Theorem.     The  successive  partial  derivatives 

d^z        d^z 

,    

dy  dx    dx  dy 
are  equal  for  any  values  of  x  and  y  in  the  vicinity  of  which 
z  and  its  first  and  second  partial  x-  and  ^-derivatives  are 
continuous. 

For,  in  z  =  f{x,  y),  first  change  x  into  a;  +  A,  keeping  y 


122  DIFFERENTIAL   CALCULUS  [Ch.  VIII. 

constant.  Then  by  the  theorem  of  mean  value  (Art.  45), 
the  increment  of  the  function  is  equal  to  the  increment  of 
the  variable  multiplied  by  the  derivative  taken  for  some 
value  intermediate  between  x  and  x-{-h\  that  is, 

fQc  +  h,  y)  -f(x,  y^=h  ±f(x  +  ^A,  ?/) .      [0  <  ^  <  1. 

Next  let  y  change  to  y  +  k^  x  remaining  constant,  and 
take  the  increment  of  the  function  on  the  left.  Then  by 
the  theorem  of  mean  value  applied  to  — f(x  +  dh^  y)  as  a 
function  of  y  with  the  increment  ^, 

lf(x  +  li,y  +  h) -fix,  y  +  m- if(x  +  h  y) -fix,  y)] 

=  kh4-^f(x  +  eh,y  +  e^k-). 

dy  ax 
Now  let  these  increments  be  given  in  reversed  order.    Then 
[/(^  +  h,y  +  1c)-f(ix  +  h,  y)]  -  IfCx,  y  +  k^  -fix,  y)] 

^^k^-^f(x-\-eji,y{.ejcy, 

dx  dy 
hence 

i.  ±fQc  +  eh,  y  +  e,k-)=  i-  -^fCx  +  e,h,  y  +  ejc). 

dy  ox  dx  dy 

This  relation  is  true  for  any  values  of  h  and  k  for  which 
all  the  functions  mentioned  are  continuous. 
When  h,  k  approach  zero, 

x-\-  Oh,  y  +  B^k,  and  x  +  6Ji^  y  +  6jc 
approach  a;,  y,  and 

fix  +  eh,y  +  e^k),  f(x  +  e^h,  y  +  ejc^ 

approach  /(a;,  y)\  and  similarly  for  the  derivatives;  hence 

Tyl-J^-yH.Ty^^^^y^^ 

or,  since /(a;,  y)  =  2, 

d^z   ^    ^z 
dy  dx     dx  dy 


73.]  FUNCTIONS   OF  TWO   VARIABLES  123 

Cor.  It  follows  directly  that  under  corresponding  con- 
ditions the  order  of  differentiation  in  the  higher  partial 
derivatives  is  indifferent. 

dh         dh        a% 


Kg., 


dx  by  dx      dx^  by      dy  dx^ 


EXERCISES 

^1.  Verify  that  -^  =  -^,  when  u  =  xY- 
dx  dy     oy  ox 

y    2.   Verify  that  ^^  =  ^^,  when  w  =  a:2z/  +  a:j^8 

^    3.   Verify  that  -^- -^,  when  w  =  y  log  (1  +  :^^). 
dx  dy     dy  dx 

^     4.   In  Ex.  3  are  there  any  exceptional  values  of  x,  y  for  which  the 
relation  is  not  true  ? 

V-  5.   Given  m  =  (x^  +  y^p,  verify  the  formula 

dx"^  dx  dy         dy^ 

/  6.   Given  m  =  (x^  +  3/^)^,  show  that  the  expression  in  the  left  member 
of  the  differential  equation  in  Ex.  5  is  equal  to  — — 

/     7.   Given  u  =(x'^-\- y'^  +  s^)-^ .   prove  that  |!^  +  ^  +  |!^  =  0. 

dx^      dy^      dz^ 

8.  Given  m  =  sec  (v  +  ax)  +  tan  Cy  —-ax)  ;    prove  that  -—  =  a^—-- 

dx^  dy^ 

9.  Given  m  =  sm  a:  cos  y ;  verify  that  =  :,:,:,:,    =  TTK~i' 

dy^  dx'^      dx  dy  dx  dy      dx^  dy^ 

^  10.  Given  u  =  (4  a6  -  c^)"^ ;  prove  that  |^'  =  ^^. 


CHAPTER   IX 

CHANGE   OF   VARIABLE 

74.  Interchange  of  dependent  and  independent  variables. 
If  ^  be  a  continuous  function  of  a;,  defined  by  the  equation 

/(a?,  y)  =  0,  the  symbol  -^  represents  the  derivative  of  y 

dx 

with  regard  to  x^  when  one  exists.     If  a;  be  regarded  as  a 

function  of  y,  defined  by  the  same  equation,  the  symbol  — 

dy 

represents  the  derivative  of  x  with  regard  to  y,  when  one 
exists.      It   is  required    to   find    the    relation  between  -^ 

and  ^.  •"" 

dy 

Let  x^  y  change  from  the  initial  values  x^^  y^  to  the  values 

x^  H-  Aa;,  y^  +  Ay,  subject  to  the  relation  /(a;,  y)  =  0. 

Then,  since 

Aa:      Aa;' 
Ay 

it  follows,  by  taking  the  limit,  that 

dx     dx  ^^ 

dy 

Hence,  if  y  and  x  he  connected  hy  a  functional  relation  the 
derivative  of  y  with  regard  to  x  is  the  reciprocal  of  the  derivative 
of  X  tvith  regard  to  y. 

This  process  is  known  as  changing  the  independent  varia- 
ble from  X  to  y.     The  corresponding  relations  for  the  higher 

124 


Ch.  IX.  74-75.] 


CHANGE  OF   VARIABLE 


125 


derivatives  are  less  simple.     They  are  obtained  in  the  follow- 
ing manner : 

To  express  —4  in  terms  of  — -,    — -r,  differentiate  (1)  as 
dx^  ay     dy^ 

to  a;;  ^       if 


^_d^ 
dx^      dx 


fl  1 

d 

fi-i 

dx 

dy 

dx 

idy\ 

[dy} 

dy       d   {  1  ^ 


dx      dy 


dx 
Vdy} 


dx 


But 


hence 


dJh 

d^  rj_1  _        dy'^ 

dx\^' 


dy 


dx 
<dy\ 


In  a  similar  manner, 


d^ 


d7? 


dxV 

dy) 

dy\ 
fdxS^ 
\dy) 

d^x  dx  _  o  fd^x^ 
dy^  dy         Wy V  ^ 

dx\^ 

dy. 


(2) 


(3) 


75.  Change  of  the  dependent  variable,     li  y  is  sl  function 
of  25,  let  it  be  required  to  express  -^,    — ^,  •••  in   terms   of 

^,  ^, .... 

dx     dx^ 

Suppose  y  =  (f>(z).     Then 

dy_^dydz^^Mt(^^^dz^ 
dx      dz  dx  dx 


dx^      dx 


(^'^43 


126  DIFFERENTIAL   CALCULUS  [Ch.  IX. 

g  =  r(.)(|)V^'(.)S.  (4) 

The  higher  a;-derivatives  of  y  can  be  similarly  expressed 
in  terms  of  a:-derivatives  of  z. 

76.   Change    of   the    independent  variable.      Let   t^  be  a 
function  of  x^  and  let  both  x  and  y  be  functions  of  a  new 

variable  t.     It  is  required  to  express  -^  in  terms  of   -^, 

TO  1  -n 

and  -T~  in  terms  of  -^  and  -=4* 
do!^  dt  dt^ 


By  Art.  14, 


dy__  dt 
dx  ~  dx^ 

dt 

d^y  dx 
cPy      dt^  dt 

dh;d]i 
dt^  dt 

dx^  ~           fd 

x\^ 

(1) 


hence  u^      u^^_u^ — u^i_u^^  ^2) 


In  practical  examples  it  is  usually  better  to  work  by  the 
methods  here  illustrated  than  to  use  the  resulting  formulas. 

EXERCISES 

1.   Change  the  independent  variable  from  a:  to  2  in  the  equation 


A. 


^»+^l  +  2'  =  '''  ''^""   '  =  *•• 


dy 
dz 


dy 

dx~ 

dz 

dH. 

Hence  x2^,  +  z^  +  y  =  0  becomes  ^i  +  y  =  0. 


7r>-76.]  CHANGE  OF  VARIABLE  127 

*    2.   Interchange  the  function  and  the  variable  in  the  equation 

/    3.  Interchange  x  and  y  in  the  equation 


R 


'      4.   Change  the  independent  variable  from  a:  to  ^  in  the  equation 

Jd^y  _dl  fu  _d^fdyy^  ^^ 
\dx^/       dx  dx^      dx'^XdxJ 

*^  5.   Change  the  dependent  variable  from  y  to  2  in  the  equation 

d^y      ,       2n+y)fdyy       ^ 

/    6.   Change  the  independent  variable  from  x  to  y  in  the  equation 

x^ \-  X 1-  M  =  0,  when  y  =  log  x. 

dx^         dx 

7.   If  y  is  a  function  of  x,  and  x  a  function  of  the  time  t,  express  the 
^-acceleration  in  terms  of  the  a:-acceleration,  and  the  x-velocity. 

Since  dy^dy^Ix^ 

dt      dx  dt 

hence  d^  ^d_yd^^lx  ,  ^dJJ\ 

dt^      dx  dt^      dt    dAdxJ 

But  ^(^Ie\  =  jL(^]^I^  =  ^^j 

dt\dxJ      dxKdxl  dt     dx'^  dt 

hence  d^^dj,d^     d^nix\\ 

dt^      dx  dt^      dxAdt) 

In  the  abbreviated  notation  for  ^derivatives, 

«4.   8.   Change  the  independent  variable  from  x  to  u  in  the  equation 

d'^y         2  X     dy  y  ^1 

-t4  +  ^ 5  -r-  +  ,,         0x0  =  0,   when   x  =  tan  u. 

dx^      1  -\-  x2  t/a;      (1  +  x^y 


128  DIFFERENTIAL   CALCULUS  [Ch.  IX.  76. 

^     9.   Change  the  independent  variable  from  x  to  f  in  the  equation 
^^-'^'^S-'^i^^'  ""^^^   :r  =  cos^ 
10.   Show  that  the  equation 

remains  unchanged  in  form  by  the  substitution  2:  =  -• 

r  11.  Interchange  the  variable  and  the  function  in  the  equation 

dx^  \dxi   y\dxi   " 

^      12.  Change  the  dependent  variable  from  y  to  2  in  the  equation 


APPLICATIONS   TO    GEOMETRY 


CHAPTER   X 
TANGENTS  AND  NORMALS 

77.  It  was  shown  in  Art.  10  that  if -/(a;,  «/)  =  0  be  the 

dy 
equation  of  a  plane  curve,  then  -~  measures  the  slope  of  the 

tangent  to  the  curve  at  the  point  x,  y.  The  slope  at  a  partic- 
ular point  (a^j,  ^j)  will  be  denoted  by  -~=^  meaning  that  x^  is 
to  be  substituted  for  x^  and  y^  for  y  in  the  expression  for  -^. 

(XX 

78.  Equation   of  tangent  and   normal   at  a  given   point. 
Since  the  tangent  line  passes  through  the  given  point  (a^j,  y-[) 

and  has  the  slope  -^,  its  equation  is 

The  normal  to  the  curve  at  the  point  (x^^  y{)  is  the  straight 
line  through  this  point,  perpendicular  to  the  tangent. 
Since  the  slope  of  the  normal  is 

:zl=_^,  [Art.  74, 

dy  dy 

dx 


(2) 


its  equation  is 

dx-,  r                ^ 

y      y,=      ^(^      x,y 

i.e.. 

<^^-^i)+ii^2'-2'>>= 

129 

130 


DIFFERENTIAL   CALCULUS 


[Ch.  X. 


79.  Length  of  tangent,  normal,  subtangent,  subnormal. 

The  segments  of  the  tangent  and  normal  intercepted  be- 
tween the  point  of  tangency  and  the  axis  OX  are  called, 
respectively,  the  tangent  length  and  the  normal  length, 
and  their  projections  on  OX  are  called  the  subtangent  and 
the  subnormal. 


Fig.  23  a. 


Fig.  23  h. 


Thus,  in  Fig.  23,  let  the  tangent  and  normal  to  the  curve 
P(7  at  P  meet  the  axis  OX  in  T  and  N,  and  let  MP  be  the 
ordinate  of  P.     Then 

TP  is  the  tangent  length, 
PN  the  normal  length, 
TM  the  subtangent, 
JOT  the  subnormal. 

These  will  be  denoted,  respectively,  by  t,  w,  t,  v. 

Let  the  angle  XTP  be  denoted  by  <^,  and  write  tan<^=m. 

1 


Then 


cos<^  = 


vr+ 


;  sin<^  = 


m 


m" 


VI  + 


m'^ 


t  =  -^  =  ^ — ;    w  =  -^=yiVl  H-TTi^; 

sin  9  m  cos  9 


T  =  j,,cot^  =  y,^^J=^;  r  =  y,  tan <^=«/,^=  my,. 


79.]  TANGENTS  AND  NORMALS  131 

The  subtangent  is  measured  from  the  intersection  of  the 
tangent  to  the  foot  of  the  ordinate ;  it  is  therefore  positive 
when  the  foot  of  the  ordinate  is  to  the  right  of  the  intersec- 
tion of  tangent.  The  subnormal  is  measured  from  the  foot 
of  the  ordinate  to  the  intersection  of  normal,  and  is  positive 
when  the  normal  cuts  OX  to  the  right  of  the  foot  of  the 
ordinate.  Both  are  therefore  positive  or  negative,  according 
as  (f)  is  acute  or  obtuse. 

The  expressions  for  t,  v  may  also  be  obtained  by  finding 
from  equations  (1),  (2),  Art.  78,  the  intercepts  made  by  the 
tangent  and  normal  on  the  axis  OX.  The  intercept  of  the 
tangent  subtracted  from  x-^^  gives  r,  and  x-^^  subtracted  from 
the  intercept  of  the  normal  gives  v. 

Ex.  Find  the  intercepts  made  upon  the  axes  by  the  tangent  at  the 
point  (x^,  ?/j)  on  the  curve  y/x  +  Vy  =  -\/a,  and  show  that  their  sum  is 
constant. 

Differentiating  the  equation  of  the  curve, 

y/x      y/y  ^^ 
Hence  the  equation  of  the  tangent  is 


The  X  intercept  is  x^  +  V^^p  and  the  y  intercept  is  y^  +  Vx^~y[,  hence 
their  sum  is 

If  a  series  of  lines  be  drawn  such  that  the  snm  of  the  intercepts  of 
each  is  the  same  constant,  account  being  taken  of  the  signs,  the  form 
of  the  parabola  to  which  they  are  all  tangent  can  be  readily  seen. 

EXERCISES 

1.  Find  the  equations  of  the  tangent  and  the  normal  to  the  ellipse 
^  +  ^  =  1  at  the  point  (xp  y^) .  Compare  the  process  with  that  employ*^.d 
in  analytic  geonietry  to  obtain  the  same  results. 


132  DIFFERENTIAL   CALCULUS  [Ch.  X. 

2.   Find  the  equation  of  the  tangent  to  the  curve  x^{x-\-y)=a\x—y) 
at  the  origin. 

/     3.   Find  the  equations  of  the  tangent  and  normal  at  the  point  (1,  3) 
on  the  curve  y^  =  9  x^. 

y  4.   Find  the  equations  of  the  tangent  and  normal  to  each  of  the 
following  curves  at  the  point  indicated: 

(a)  V  = )  at  the  point  for  which  a:  =  2  a. 

(^)  y2  _  2  a:2  —  z*,  at  the  points  for  which  x  =  1. 
(y)  ?/2  _  ^py.^  at  the  point  (/>,  2p). 

5.   Find  the  value  of  the   subtangent  of    y'^  =  ^x'^—12   at   a;  =  4. 
Compare  the  process  with  that  already  given  in  analytic  geometry. 

SI     6.   Find  the  length  of  the  tangent  to  the  curve  ^y^  —  2  a:  at  a;  =  8. 

~^  7.  Find  the  points  at  which  the  tangent  is  parallel  to  the  axis  of  a:, 
and  at  which  it  is  perpendicular  to  the  x  axis  for  each  of  the  following 
curves : 

(a)    ax^-\-2hxy+hy'^  =  l. 

08)  y  = 


ax 
(y)  y^  =  x\2a-x). 

-w     8.   Find  the  condition  that  the  conies  ax'^  +  hy^=l,  a'x^  +  l/y^=  1 
shall  cut  at  right  angles. 

T      9.   Find  the  angle  at  which  x^  =  y^  +  5  intersects  Sx^-\-lSy^=  144. 
Compare  with  Ex.  8. 

10.  Show  that  in  the  equilateral  hyperbola  2  xy  =  a^  the  area  of  the 
triangle  formed  by  a  variable  tangent  and  the  coordinate  axes  is  constant 
and  equal  to  a^. 

11.  At  what  angle  does  y^  =  Sx  intersect  4 a:*  +  2 y^  =  43 ? 

12.  Determine  the  subnormal  to  the  curve  y»  =  a*-^  x. 

13.  Find  the  values  of  x  for  which  the  tangent  to  the  curve 

?/8  =  (x-a)2(x-c) 
is  parallel  to  the  axis  of  x. 

14.  Show  that  the  subtangent  of  the  hyperbola  xy  =  a^  is  equal  to 
the  abscissa  of  the  point  of  tangency,  but  opposite  in  sign. 

/    15.  Prove  that  the  parabola  y*  =  4  aa:  has  a  constant  subnormal. 


79-80.] 


TANGENTS  AND  NORMALS 


133 


16.  Show  analytically  that  in  the  curve  x^  +  y^  =  a^  the  length  of  the 
normal  is  constant. 

17.  Show  that  in  the  tractrix,  the  length  of  the  tangent  is  constant, 
the  equation  of  the  tractrix  being 

2     ^e  +  V^^37 


18.  Show  that  the  exponential  curve  y  =  ae"  has  a  constant  sub- 
tangent. 

19.  Find  the  point  on  the  parabola  y^  =  4:px  at  \v^hich  the  angle 
between  the  tangent  and  the  line  joining  the  point  to  the  vertex  shall  be 
a  maximum. 

POLAR   COORDINATES 

80.  When  the  equation  of  a  curve  is  expressed  in  polar 
coordinates,  the  vectorial  angle  6  is  usually  regarded  as  the 
independent  variable.  To  determine  the  direction  of  the 
curve  at  any  point,  it  is  most  convenient  to  make  use  of 
the  angle  between  the  tangent  and  the  radius  vector  to  the 
point  of  tangency. 

Let  P,  Q  be  two  points  on  the 
curve  (Fig.  24).  Join  P,  Q  with 
the  pole  0,  and  drop  a  perpendic- 
ular PM  from  P  on  OQ.  Let  /?, 
0  be  the  coordinates  of  P;  p+Ap^ 
6+AO  those  of  Q.  Then  the  angle 
P0$  =  A(9;  Pil[/  =  /3sinA6>;  and 
MQ=OQ-OM=p  +  Ap-p  cos  Ad. 


Fig.  24. 


Hence 


tan  MQP  = 


p  sin  A^ 


p  +  Ap  —  p  cos  A^ 

When  Q  moves  to  coincidence  with  P,  the  angle  MQP 
approaches  as  a  limit  the  angle  between  the  radius  vector 
and  the  tangent  line  at  the  point  P.  This  angle  will  be 
designated  by  yjr. 


DIFFERENTIAL   CALCULUS 


lira 


134 
Thus 
But    p(l  -  cos  Ae)=2p  sin2 1 A^, 


[Ch.  X. 


t^^r-A^-Op  +  Ap_pcosA^ 


hence 


^^^i^  =  Jlo 


p  sin  A^ 
A(9 


.    1  ./J     siniA^  ,  Ap 


Since  .  l^^  ^  ^^"  ^       =  i^  the  preceding  equation  reduces  to 

^       ,        p  dO 

tant  =  ^=P^-. 

dd 

Ex.  1.  A  point  describes  a  circle  of  radius  p. 
Prove  that  at  any  instant  the  arc  velocity  is  p  times 
the  angle  velocity, 

(It      P  dt 


dt 


Fia.2t). 


Fio.  27. 


Ex.  2.    When  a  point  describes  a  given 
curve,  prove  that  at  any  instant  the  velocity 

^    —  has  a  radius  component  -^  and  a  com- 
''*      dt  ^  dt 

ponent  perpendicular  to  the  radius  vector 

p  — ,  and  hence  that 
dt 

C08i^  =  ^,  8mtf/  =  p^,  tan.Zr  =  p^. 
ds  ds  dp 

81.  Relation  between  ^  and  ^• 
dx  dp 

If  the  initial  line  be  taken  as  the  axis 
of  X,  the  tangent  line  at  P  makes  an 
angle  (f)  with  this  line. 

Hence         6  +  yjr  =  ^; 


80-82.] 


TANGENTS  AND  NOBMALS 


135 


82.  Length  of  tangent,  normal,  polar  subtangent,  and  polar 
subnormal.  The  portions  of  the  tangent  and  normal  inter- 
cepted between  the  point  of  tangency  P  and  the  line  through 
the  pole  perpendicular  to  the  radius  vector  OP,  are  called 
the  polar  tangent  length  and  the  polar  normal  length; 
their  projections  on  this  perpendicular  are  called  the  polar 
subtangent  'dudi  polar  subnormal. 


Fig.  28  a.  Fig.  28  6. 

Thus,  let  the  tangent  and  normal  at  P  meet  the  perpen  • 
dicular  to  OP  in  the  points  JV  and  M.     Then 

PN  is  the  polar  tangent  length, 
PM  is  the  polar  normal  length, 
ON  is  the  polar  subtangent, 
OMi^  the  polar  subnormal. 

They  are  all  seen  to  be  independent  of  the  direction  of 
the  initial  line.  The  lengths  of  these  lines  will  now  be 
determined.  

Since  PN=  OP .  sec  OPN=  psec^jr^  pyjp^f^J  +  1 


-'f.M%i 


dp 


hence    polar  tangent  length  =  p  -^\P^  +  ( -^ 


dO 


136  DIFFERENTIAL   CALCULUS  [Ch.  X. 

Again,  0N=  OP  tan  OPN=  /o  tan  -f  =  p2  ^, 

dp 

hence  polar  subtangent  = /o^ -—• 

dp 


PM=  OP  '  CSC  (9PiV^=  pGSGylr=  \p^  +  f^\ 
hence      polar  normal  length  =  \p^  "*■  V;^)  * 
0M=  OP  cot  OP]Sr=  ^, 

hence  polar  subnormal  =  -^* 

The  signs  of  the  polar  tangent  length  and  polar  normal 
length  are  ambiguous  on  account  of  the  radical.     The  direc- 

tion  of  the  subtangent  is  determined  by  the  sign  of  p^ — . 

dS  ^P 

When  --  is  positive,  the  distance  ON^  should  be  measured 
dp 

to  the  right,  and  when  negative,  to  the  left  of  an  observer 
placed  at  0  and  looking  along  OP;    for  when  0  increases 

with  p,  —-  is  positive  (Art.  13),  and  ylr  is  an  acute  angle  (as 

dp  ^Q 

in  Fig.  28  h) ;  when  0  decreases  as  p  increases,  —  is  negative, 

and  i/r  is  obtuse  (Fig.  28  a).  ^ 

EXERCISES 

1.  In  the  curve  p  =  a  sin  ^,  find  \p. 

2.  In  the  spiral  of  Archimedes  p  =  a$f  show  that  tan  \^  =  0  and  find 
the  polar  subtangent,  polar  normal,  and  polar  subnormal.  Trace  the 
curve. 

3.  Find  for  the  curve  p^  =  a*co8  2d  the  values  of  all  the  expressions 
treated  in  this  article. 

4.  Show  that  in  the  curve  pO  =  a  the  polar  subtangent  is  of  constant 
length.    Trace  the  curve. 


82.]  TANGENTS  AND  NORMALS  187 

5.  In  the  curve  p=a(l  —  cosO),  find  i/r  and  the  polar  subtangent. 

6.  Show  that  in  the  curve  p  =  b  •  e^cota  the  tangent  makes  a  constant 
angle  a  with  the  radius  vector.  For  this  reason,  this  curve  is  called  the 
equiangular  spiral. 

yC  7.   Find  the  angle  of  intersection  of  the  curves 

p  =  a(l  +  cos  6),  p  =  6(1  —  cos^. 

8.  In  the  parabola  p  =  a  sec^  -,  show  that  ^  +  ^  =  w. 


CHAPTER   XI 

DERIVATIVE  OF   AN   ARC,  AREA,  VOLUME,   AND   SURFACE 
OF  REVOLUTION 

83.  Derivative  of  an  arc.     The  length  s  of  the  arc  AP  of 

a  given  curve  1/  =/(a:),  measured  from  a  fixed  point  A  to  any 

point  P,  is  a  function  of  the  abscissa  x  of  the  latter  point, 

and  may  be  expressed  by  a  relation  of  the  form  s  =  (K^^)- 

The  determination  of   the  function  (f>  when  the  form  of 

/  is  known,  is  an  important  and  sometimes  difficult  problem 

in  the  Integral  Calculus.     The  first  step  in  its  solution  is 

ds 
to  determine  the  form  of  the  derivative  function  -—  =  <t>'(x)^ 

ax 

which   is  easily  done  by  the  methods  of  the   Differential 

Calculus. 

Let  PQ  \)Q  two  points  on  the  curve  (Fig.  29);    let  x^  y 

be  the  coordinates  of  P ;  a:  +  Aa:, 

y  -\-  ^y  those  oi  Q\  s  the  length 

of  the  arc  AP ;    s  +  A«  that  of 

the  arc  AQ.     Draw  the  ordinates 

MP,  NQ  ;  and  draw  PR  parallel 

— 2C_      to  MN.  Then  PR= Ax,  RQ=  Ay; 


M     N 

J.IQ29.  arcP^=As.     Hence 


Chord  PQ  =  V(Aa:)2  +  (Ay)2, 


Ax 


T..„,.„=.^.^.|iV..(g)' 


As  _  As 

Ax     PQ     Ax      PQ 
138 


Ch.  XI.  83-84.]       DERIVATIVES   OF  ARC,   AREA,   ETC. 


139 


Taking  the  limit  of  both  members  as  Ax  approaches  zero 
and  putting  ^^"Iq-^^^  1,  by  Art.  6,  Th.  4,  and  Art.  4, 
Th.  8,  Cor.,  it  follows  that 


%'Mf)'- 


Similarly 

Moreover,  from  Art.  65 

or  in  the  differential  notation 


dx^      fdy^ 
dt  \dt 


)'■ 


(1) 

(2) 

(3) 
(4) 


84.   Trigonometric  meaning  of 


ds     ds 
dx    dy 


Since 


PQ 


As      PQ     As 
it  follows  by  taking  the  limit  that 

dx  , 

-—  =  cos  9, 


wherein  <j),  being  the  limit  of  the  angle  RPQ,  is  the  angle 
which  the  tangent  at  the  point  (x,  y)  makes  with  the  a;-axis. 

Similarly,  -^  =  sin  <^ ;   whence  — -  =  sec  <^,  — -  =  esc  </>. 


ds       -   - r  ^  -  ^^ 

Using  the  idea  of  a  rate  or  dif- 
ferential, all  these  relations  may 
be  conveniently  exhibited  by  Fig. 
30. 

These  results  may  also  be  de- 
rived from  equations  (1),  (2)  of 

Art.  83,  by  putting  ^  =  tan  </>. 


dy 


Y 

dy 

y. 

I- 

^1 

0 

X 

Fia.  30. 


140 


DIFFERENTIA  L   CALCUL US 


[Ch.  XI. 


85.  Derivative  of  the  volume  of  a  solid  of  revolution.     Let 
the  curve  APQ  revolve  about  the  a:-axis,  and  thus  generate 

a  surface  of  revolution  ;  let  V 
be  the  volume  included  between 
this  surface,  the  plane  generated 
by  the  fixed  ordinate  at  A,  and 
the  plane  generated  by  any  ordi- 
nate MP. 

Let  A I^  be  the  volume  gener- 
ated by  the  area  PMNQ.     Then 
A  V  lies  between  the  volumes  of   the   cylinders   generated 
by  the  rectangles  PMNR  and  SMNQ;    that  is, 
iry'^a^x  <  A  r<  7r(y  -h  Ay)2Aa;. 
Dividing  by  Aa;  and  taking  limits. 


F 

• 

^ 

1 

-JL^ 

F 

r 

R 

X 

0 

A 

{ I 

V 

Fia.  30  a. 


dV 


—    m-ll^ 


86.  Derivative  of  a  surface  of  revolution.  Let  S  be  the 
area  of  the  surface  generated  by  the  arc  AP  (Fig.  31),  and 
AS  that  generated  by  the  arc  PQ  whose  length  is  As. 

Draw  PQ\  QP'  parallel  to  OX 
and  equal  in  length  to  the  arc  PQ. 
Then  it  may  be  assumed  as  an 
axiom  that  the  area  generated  by 
PQ  lies  between  the  areas  gen- 
erated by  PQ'  and  P'Q\  i.e., 

2  iryAs  <  AaS'  <  2  irQy  +  Ay) A«. 
Dividing  by  A«  and  passing  to  the  limit, 
dS 


M      N 
Fio.  31. 


!;-'-'>• 


f-f-l— V^^ 


(1) 
(2) 


85-87.]  DERIVATIVES   OF  ARC,   AREA,   ETC.  141 

87.  Derivative  of  arc  in  polar  coordinates. 

Let  /3,  6  be  the  coordinates  of  P ;  p  -\-  A/a,  6  +  A^  those 
oi  Q  ;  s  the  length  of  the  arc  KP ; 
As  that  of  arc  PQ ;   draw  Pilif  per- 
pendicular to  OQ.     Then 

PM=p  sin  AO, 

MQ=OQ-OM==p^Ap-p  cos Ae 

=p(l  —  GosA6)  +  Ap  o^ 

=  2/)sin2iA<9-hA/). 
Hence    PQ^=Cp  sin  Al9)2  -f  (2  /o  sin2  J  A(9  +  A/o)2, 

Replacing  the  first  member  by  ( — -^  *  t4  ) '  passing  to  the 

\  As     Ad  J 

limit  when  A^  =  0,  and  putting  lim  — ^  =  1,  lim  ^^^     -  =  1, 

1  A/?  ^^  ^^ 

lim^i^4^=  1,  it  follows  that 
^Ad 


Fig.  32. 


©■='■- gj 


*•'•'  d0 


Mtl 


In  the  rate  or  differential  notation  this  formula  may  be 
conveniently  written 

d»^  =  dp^  +  p^dff^. 

This  relation  may  be  readily  deduced  also  from  Fig.  26, 
Art.  80. 


142 


DIFFERENTIAL   CALCULUS 


[Ch.  XI. 


88.  Derivative  of   area  in    polar  coordinates.     Let  A   be 

the  area  of  OKP  measured 
from  a  fixed  radius  vector  OK 
to  any  other  radius  vector  OP  ; 
^x  let  A  A  be  the  area  of  OPQ. 
Draw  arcs  PM,  QN,  with  0  as 
a  center.  Then  the  area  POQ . 
lies  between  the  areas  of  the 
sectors  OPiltf  and  ONQ\    i.e.. 


Fig.  33. 


1  /32A(9  <  A^  <  lip  +  Ap)2 A^. 

Dividing  by  A^  and  passing  to  the  limit,  when  A^  =  0,  it 
follows  that 


dA 

dd 


=  \p'- 


For  the  derivative  of  the  area  of  a  curve  in  rectangular 

dA 

coordinates,  see  Art.  10.     The  result  is  —— =  y. 

dx 


EXERCISES  ON  CHAPTER  XI 

1.  In  the  parabola  w^  =  4  ax,  find  — ,   - — ,   -— ,    — — 

dx     dx     dx     dx 

2.  Find  —  and  —   for  the  circle  x^-{-y^  =  a\ 

dx  dy  ^ 

ds 

3.  Find   —  for  the  curve  e^  cos  x  =  \. 

dx 

4.  Find  the  x-derivative  of  the  volume  of  the  cone  generated  by 
revolving  the  line  y  —  ax  about  the  axis  of  x. 

5.  Find  the  ar-derivative  of  the  volume  of  the  ellipsoid  of  revolution, 

X^        2/2 

formed  by  revolving  -^+  rj  =  1  about  its  maior  axis.      ^ 

6.  In  the  curve  p  =  a^  find  -^-  ^(ffi    "  ^ 


Im^^ 


de 


ds 


7.   Given  p  =  a(H-co8^);   find  ^. 


d$ 


8.   In  p«  =  a3  cos  2d,  find 


d$ 


c^ 


CHAPTER    XII 
ASYMPTOTES 

89.  Hyperbolic  and  parabolic  branches.  When  a  curve 
has  a  branch  extending  to  infinity,  the  tangents  drawn  at 
successive  points  of  this  branch  may  tend  to  coincide  with 
a  definite  fixed  line  as  in  the  familiar  case  of  the  hyperbola. 
On  the  other  hand,  the  successive  tangents  may  move  farther 
and  farther  out  of  the  field  as  in  the  case  of  the  parabola. 
These  two  kinds  of  infinite  branches  may  be  called  hyperbolic 
and  parabolic. 

The  character  of  each  of  the  infinite  branches  of  a  curve  can 
always  be  determined  when  the  equation  of  the  curve  is  known. 

90.  Definition  of  a  rectilinear  asymptote.  If  the  tangents 
at  successive  points  of  a  curve  approach  a  fixed  straight  line 
as  a  limiting  position  when  the  point  of  contact  moves  farther 
and  farther  along  any  infinite  branch  of  the  given  curve, 
then  the  fixed  line  is  called  an  asymptote  of  the  curve. 

This  definition  may  be  stated  more  briefly  but  less  pre- 
cisely as  follows:  An  asymptote  to  a  curve  is  a  tangent 
whose  point  of  contact  is  at  infinity,  but  which  is  not  itself 
entirely  at  infinity. 

DETERMINATION  OF   ASYMPTOTES 

91.  Method  of  limiting  intercepts.  The  equation  of  the 
tangent  at  any  point  (x^,  y{)  being 

143 


144  DIFFERENTIAL   CALCULUS  [Ch.  XII. 

the  intercepts  made  by  this  line  on  the  coordinate  axes  are 


0) 


Suppose  the  curve  has  a  branch  on  which  x==oo  and 
y  =  Qo.  Then  from  (1)  the  limits  can  be  found  to  which 
the  intercepts  rr^,  i/q  approach  as  the  coordinates  x^^  y-^  of  the 
point  of  contact  tend  to  become  infinite.  If  these  limits  be 
denoted  by  a,  5,  the  equation  of  the  corresponding  asymptote  is 

a  0 
Except  in  special  cases  this  method  is  usually  too  compli- 
cated to  be  of  practical  use  in  determining  the  equations  of 
the  asymptotes  of  a  given  curve.  There  are  three  other 
principal  methods,  which  will  always  suffice  to  determine  the 
asymptotes  of  curves  whose  equations  involve  only  algebraic 
functions.  These  may  be  called  the  methods  of  inspection, 
of  substitution,  and  of  expansion. 

92.  Method  of  inspection.  Infinite  ordinates,  asymptotes 
parallel  to  axes.  When  an  algebraic  equation  in  two  co- 
ordinates X  and  y  is  rationalized,  cleared  of  fractions,  and 
arranged  according  to  powers  of  one  of  the  coordinates,  say 
y,  it  takes  the  form 

ayn  +  (hx  4-  c)5^"-^-f  (c?r2  -f  ex  +f^y^-^+  ...  +  u„_,y  4-  w„  =  0, 

in  which  w„  is  a  polynomial  of  the  degree  n  in  terms  of  the 
other  coordinate  a;,  and  w„_i  is  of  degree  n  —  1. 

When  any  value  is  given  to  a:,  the  equation  determines  n 
values  for  y. 

Let  it  be  required  to  find  for  what  value  of  x  the  corre- 
sponding ordinate  y  has  an  infinite  value. 


91-92.]  ASYMPTOTES  145 

For  this  purpose  the  following  theorem  from  algebra  will 
be  recalled : 

Given  an  algebraic  equation  of  degree  n 

a^»  +  /3?/"-^  +  7«/""'  +  -  =  0. 

If  a  =  0,  one  root  i/  becomes  infinite  ;  if  a  =  0  and  /3  =  0, 
two  roots  1/  become  infinite ;  and  in  general  if  the  coefficients 
of  each  of  the  k  highest  powers  of  t/  vanish,  the  equation  will 
have  k  infinite  roots. 

Suppose  at  first  that  the  term  in  y"  is  present;  in  other 
words,  that  the  coefficient  a  is  not  zero.  Then,  when  any 
finite  value  is  given  to  x,  all  of  the  n  values  of  y  are  finite, 
and  there  are  accordingly  no  infinite  ordinates  for  finite 
values  of  the  abscissa. 

Next  suppose  that  a  is  zero,  and  6,  c,  not  zero.  In  this 
case  one  value  of  «/  is  infinite  for  every  finite  value  of  x^ 
and  hence  the  curve  passes  through  the  point  at  infinity 
on  the  ^  axis. 

There  is  one  particular  value  of  x,  namely,  x  =  ——,  for 

which  an  additional  root  of  the  equation  in  t/  becomes 
infinite.  For,  when  x  has  this  value,  the  coefficient  hx  +  o  oi 
the  highest  power  of  ^  remaining  in  the  equation  vanishes. 

Geometrically,  every  line  parallel  to  the  i/  axis  has  one 
point  of  intersection  with  the  curve  at  infinity,  but  the 
line    bx  +  c  =  0    has   two   points   of   intersection   with   the  i 

curve  at  infinity.     A  line  having  two  coincident  points  of  / 

intersection  with  a  curve  is  a  tangent  to  the   curve,  and     ^ 
when   the    coincident   points  are  at  infinity,    but   the   line  Ij      ^ 
itself   not  altogether  at  infinity,  the  tangent  is  an  asymp-       a 
tote.      Hence   an  ordinate  that  becomes  infinite  for  a  defi-      \    J" 
nite  value  of  x  is  an  asymptote.  ( 

Again,  if  not  only  a,  but  also  h  and  c  are  zero,  there  are 


146 


DIFFERENTIAL   CALCULUS 


[Ch.  XII. 


two  values  of  x  that  make  «/  infinite ;  namely,  those  values  of 
X  that  make  dx^-{-ex-\-f  =  0^  and  the  equations  of  the  infinite 
ordinates  are  found  by  factoring  this  last  equation  ;  and  so  on. 
Similarly,  b}^  arranging  the  equation  of  the  curve  accord- 
ing to  powers  of  a;,  it  is  easy  to  find  what  values  of  1/  give 
an  infinite  value  to  x. 

Ex.  1.   In  the  curve 

2  x^  +  x^y  +  xy^  z=  x^  -  y^  -  5, 
md  the  equation  of  the  infinite  ordinate,  and  determine  the  finite  point 
in  which  this  line  meets  the  curve. 

This  is  a  cubic  equation  in  which  the  coefficient  of  y^  is  zero. 
Arranged  in  powers  of  y  it  is 

f  (x+1)  +  yx^  +  (2  a:8  -  a:2  +  5)  =  0. 
I  the  equation  for  y  becomes 
0-1/2  +  2, +2  =  0, 

the  two  roots  of  which  are  y  =  00,  y  =  —  2 ;  hence  the  equation  of  the 
infinite  ordinate  is  x  +  JL=-0. — Xhe  infinite  ordinate  meets  the  curve 
again  in  the  finite  poin^(  —  1^  —  2J 
~^n5e~the~T«rm  m^r'Tf^^TBSBtl^  there  are  no  infinite  values  of  x  for 
finite  values  of  y. 

Ex.  2.   Show  that  the  lines  x  =  a,  and  y  =  0  are  asymptotes  to  the 
curve  a^x  =  y(x-  ay  (Fig.  34). 


¥iQ.  34. 
Ex.  3.   Find  the  asymptotes  of  the  curve  x"^  (y  -  a)  +  xy^  =  o* 


92-93.]  ASYMPTOTES  147 

93    Method   of  substitution.      Oblique   asymptotes.      The 

asymptotes  that  are  not  parallel  to  either  axis  can  be  found 
by  the  method  of  substitution,  which  is  applicable  to  all 
algebraic'  curves,  and  is  of  especial  value  when  the  equation 
is  given  in  the  implicit  form 

/(^,^)  =  0.  (1) 

Consider  the  straight  line 

y  =mz  +  b,  (2) 

and  let  it  be  required  to  determine  m  and  b  so  that  this  line 
shall  be  an  asymptote  to  the  curve  f{x,  ?/)  =  0. 

Since  an  asymptote  is  the  limiting  position  of  a  line  that 
meets  the  curve  in  two  points  that  tend  to  coincide  at 
infinity,  then,  by  making  (1)  and  (2)  simultaneous,  the 
resulting  equation  in  x, 

fix,  ma:  +  5)  =  0, 

is  to  have  two  of  its  roots  infinite.  This  requires  that  the 
coefificients  of  the  two  highest  powers  of  x  shall  vanish. 
These  coefficients,  equated  to  zero,  furnish  two  equations, 
from  which  the  required  values  of  m  and  h  can  be  deter- 
mined. These  values,  substituted  in  (2),  will  give  the 
equation  of  an  asymptote. 

Ex.  4.    Find  the  asymptotes  to  the  curve  y^  =  x^{2a  ~  x). 

In  the  first  place,  there  are  evidently  no  asymptotes  parallel  to  either 
of  the  coordinate  axes.  To  determine  the  oblique  asymptotes,  make  the 
equation  of  the  curve  simultaneous  with  y  =  mx  -\-  b,  and  eliminate  y. 
Then 

{mx-{-  by  =  x^(2a  -  x), 

or,  arranged  in  powers  of  x, 

(1  +  w8)  x^  +  (3  m%  -  2  a)  x2  +  3  b^x  +  &»  =  0. 

Let  m8  +  1  =  0    and    dm^b-  2a  =  0. 


148 


DIFFERENTIAL   CALCULUS 


[Ch.  XII. 


Then 
hence 


-a:  + 


3  ' 
2a 


3 


is  the  equation  of  an  asymptote. 

The  third  intersection  of  this  line  with  the  given  cubic  is  found  from 

the  equation  3  mb^x  +  &^  =  0,  whence  x  =  — • 
Y 


This  is  the  only  oblique  asymptote,  as  the  other  roots  of  the  equation  for 
m  are  imaginary. 

Ex.  5.   Find  the  asymptotes  to  the  curve  y  (a*  +  x^)  =  a^(a  —  x). 


Fio.  3«. 


Here  the  line  y  =  0  is  a  horizontal  asymptote  by  Art.  92.    To  find 
the  oblique  asymptotes,  put  y  =  mx  ■\-  h. 


93-94.]  ASYMPTOTES  149 

Then  (mx  +  b)  (a^  +  x^)  =  a^  (a  -  x), 

i.e.,  mx^  +  bx^  +  (ma^  +a^)x  +  {aPh  -  a^)  =  0 ; 

hence  m  =  0,    &  =  0,    for  an  asymptote. 

Thus  the  only  asymptote  is  the  line  y  =  0  already  found. 

94.  Number  of  asymptotes.  The  illustrations  of  the  last 
article  show  that  if  all  the  terms  be  present  in  the  general 
equation  of  an  nth.  degree  curve,  then  the  equation  for 
determining  m  is  of  the  nth  degree  and  there  are  accord- 
ingly n  values  of  m^  real  or  imaginary.  The  equation  for 
finding  h  is  usually  of  the  first  degree,  but  for  certain 
curves  one  or  more  values  of  m  may  cause  the  coefficient 
of  a^  and  a;""^  both  to  vanish,  irrespective  of  h.  In  such 
cases  any  line  whose  equation  is  of  the  form  y  =  m^x  +  c 
will  have  two  points  at  infinity  on  the  curve  independent 
of  c;  .but  by  equating  the  coefficient  of  a;""^  to  zero,  two 
values  of  h  can  be  found  such  that  the  resulting  lines 
have  three  points  at  infinity  in  common  with  the  curve. 
These  two  lines  are  parallel;  and  it  will  be  seen  that  in 
each  case  in  which  this  happens  the  equation  defining  m  has 
a  double  root,  so  that  the  total  number  of  asymptotes  is 
not  increased.  Hence  the  total  number  of  asymptotes,  real 
and  imaginary,  is  in  general  equal  to  the  degree  of  the 
equation  of  the  curve. 

This  number  must  be  reduced  whenever  a  curve  has  a 
parabolic  branch. 

Since  the  imaginary  values  of  m  occur  in  pairs,  it  is  evi- 
dent that  a  curve  of  odd  degree  has  an  odd  number  of  real 
asymptotes ;  and  that  a  curve  of  even  degree  has  either  no 
real  asymptotes  or  an  even  number.  Thus,  a  cubic  curve 
has  either  one  real  asymptote  or  three ;  a  conic  has  either 
two  real  asymptotes  or  none. 


150  DIFFERENTIAL   CALCULUS  [Ch.  XII. 

95.  Method  of  expansion.  Explicit  functions.  Although 
the  two  foregoing  methods  are  in  all  cases  sufficient  to  find 
the  asymptotes  of  algebraic  curves,  yet  in  certain  special 
cases  the  oblique  asymptotes  are  most  conveniently  found 
by  the  method  of  expansion  in  descending  powers.  It  is 
based  on  the  principle  that  a  straight  line  will  be  an  asymp- 
tote to  a  curve  when  the  difference  between  the  ordinates 
of  the  curve  and  of  the  line,  corresponding  to  a  common 
abscissa,  approaches  zero  as  the  abscissa  becomes  infinite. 

It  will  appear  from  the  process  of  applying  this  principle 
that  a  line  answering  the  condition  just  stated  will  also 
satisfy  the  original  definition  of  an  asymptote. 

The  principal  value  of  the  method  of  expansion  is  that 
it  exhibits  the  manner  in  which  each  infinite  branch  ap- 
proaches its  asymptote. 

Ex.   Find  the  asymptotes  of  the  curve 
a:—  3 

TT  n  \  X/\  Xl 

Here  y^  = :r^^ , 

Hence  the  oblique  asymptotes  are  y  =  ±{x  —  1)  (Fig.  37). 
The  sign  of  the  next  term  shows  that  when  z  =  +  x),  tlie  curve  is  above 
the  first  asymptote  and  below  the  second;  and  vice  versa  when  a;  ==  —  oo. 


95.] 


ASYMPTOTES 


151 


The  same  method  may  be  applied  to  cases  in  which  x  is  an  explicit 
function  of  y. 

This  method  can  also  be  extended  so  as  to  apply  to  curves  defined  by 
an  implicit  equation,  f(x,  y)  =  0.  [See  McMahon  and  Snyder's  "  Differ- 
ential Calculus,"  p.  234.] 


Fig.  37. 


EXERCISES  ON  CHAPTER  XII 
Find  the  asymptotes  of  each  of  the  following  curves : 
1.  y(a^  -  a;2)  =  b(2  x -\-  c).  ■'    7.    (x  +  a)f  =  (y  +  h)x\ 


8.   x' 


x^+  X  +  y. 


\f 


2      2^q^(a:-a)(a:-3a). 
■  ^  x'^-2ax 

3.   a:Y  ^  a\x'^  -  y^). 
^    4.  y  =  a+  . 

(X  -  C)2 

/  5.   /  =  x%a  -  x). 
^  6.   y\x-l)  =  x^. 

15.  x^  +  2x^y  -  xy^-2y^  +  4:y^  +  2xy  +  y=l. 


9.  xy^  +  x^y  =  a^. 

10.  2/(^2  +  3  a2)  =  a;8. 

A  11.  a:3  -  3  aar?/  +  ^8  _  q. 

12.  x3  +  ?/8  =  aS. 

13.  x4  -  a;  V  +  a'^^^  +  6*  =  0. 


CHAPTER   XIII 

DIRECTION  OF  BENDING.    POINTS  OF  INFLEXION 

96.  Concavity  upward  and  downward.  A  curve  is  said  to 
be  concave  downward  in  the  vicinity  of  a  point  P  when, 
for  a  finite  distance  on  each  side  of  P,  the  curve  is  situated 


below  the  tangent  drawn  at  that  point,  as  in  the  arcs  -42), 
FH.  It  is  concave  upward  when  the  curve  lies  above  the 
tangent,  as  in  the  arcs  DF^  HK. 

By  drawing  successive  tangents  to  the  curve,  as  in  the 
figure,  it  is  easily  seen  that  if  the  point  of  contact  advances 
to  the  right,  the  tangent  swings  in  the  positive  direction  of 
rotation  when  the  concavity  is  upward,  and  in  the  negative 
direction  when  the  concavity  is  downward.  Hence  upward 
concavity  may  be  called  a  positive  bending  of  the  curve,  and 
downward  concavity,  negative  bending. 

A  point  at  which  the  direction  of  bending  changes  con- 
tinuously from  positive  to  negative,  or  vice  versa,  as  at  F  oi 

102 


Ch.  XIII.  96-97.]       DIRECTION  OF  BENDING  163 

at  D,  is  called  a  point  of  inflexion^  and  the  tangent  at  such 
a  point  is  called  a  stationary  tangent. 

The  points  of  the  curve  that  are  situated  just  before  and 
just  after  the  point  of  inflexion  are  thus  on  opposite  sides  of 
the  stationary  tangent,  and  hence  the  tangent  crosses  the 
curve,  as  at  i>,  #,  H. 

97.  Algebraic  test  for  positive  and  negative  bending.     Let 

the  inclination  of  the  tangent  line,  measured  from  the  right- 
hand  end  of  the  a;-axis  toward  the  forward  (right-hand)  end 
of  the  tangent,  be  denoted  by  </>.  Then  <f>  is  an  increasing 
or  decreasing  function  of  the  abscissa  according  as  the  bend- 
ing is  positive  or  negative ;  for  instance,  in  the  arc  AD^  the 
angle  <f>  diminishes  from  +  —  through  zero  to  —  — ;   in  the 

arc  i>jP,  <f)  increases  from  —  -j  through  zero  to  — ;  in  the  arc 

FH^  (f)  decreases  from  -f  —  through  zero  to  —  —  J  ^^^  i^  *^® 

arc  HK^  </>  increases  from  —  —  through  zero  to  +  — • 

2i  4 

At  a  point  of  inflexion  </>  has  evidently  a  turning  value 
which  is  a  maximum  or  minimum,  according  as  the  concavity 
changes  from  upward  to  downward,  or  conversely. 

Thus  in  Fig.  38,  <^  is  a  maximum  at  JP,  and  a  minimum  at 
D  and  at  S, 

Instead  of  recording  the  variation  of  the  angle  </>,  it  is 

generally  convenient  to  consider  the  variation  of  the  slope 

tan<^,  w^hich  is  easily  expressed  as  a  function  of  x  by  the 

equation 

tan  <f)  =  -^• 
^      ax 

Since  tan  <^  is  always  an  increasing  function  of  <^,  it  follows 

that  the  slope  function  -=^  is  an  increasing  or  a  decreasing 

ax 


154  DIFFERENTIAL   CALCULUS  [Ch.  XIII. 

function  of  a;,  according  as  the  concavity  is  upward  or  down- 
ward, and  hence  that  its  a;-derivative  is  positive  or  negative. 
Thus  the  bending  of  the  curve  is  in  l^e  positive  or  nega- 

tive  direction  of  rotation,  according  as  the  function  -t4  is 

positive  or  negative. 

dii 
At  a  point  of  inflexion  the  slope  -^  is  a  maximum  or 

minimum,  and  therefore  its  derivative  -t4  changes  sign  from 

positive  to  negative  or  from  negative  to  positive.  This 
latter  condition  is  evidently  both  necessary  and  sufficient  in 
order  that  the  point  (x^  y)  may  be  a  point  of  inflexion  on 
the  given  curve. 

Hence,  the  coordinates  of  the  points  of  inflexion  on  the 
curve  y^fix) 

may  be  found  by  solving  the  equations 

and  then  testing  whether  f'Qx)  changes  its  sign  as  x  passes 
through  the  critical  values  thus  obtained.  To  any  critical 
value  a  that  satisfies  the  test,  corresponds  the  point  of 
inflexion  (a^f(a)'). 

Ex.  1.   For  the  curve 

find  the  points  of  inflexion,  and  show  the  mode  of  variation  of  the  slope 
and  of  the  ordinate. 

Here  i  =  *^(^"-l)' 

g  =  4(3.-l). 

hence  the  critical  values  for  inflexions  are  ar  =  ±  — .    It  will  be  seen 

1  ^ 

that  as  a;  increases  through  — =-,  the  second  derivative  changes  sign  from 

\/3 

positive  to  negative,  hence  there  is  an  inflexion  at  which  the  concavity 

changes  from  upward  to  downward.    Similarly,  at  x  =  +  — ^  the  con- 

V3 


97.] 


DIRECTION  OF  BENDING 


155 


X 

y 

dy 
dx 

—  CO 

+  00 

—  c» 

+ 

_  2 

+  25 

-24 

+ 

-1 

0 

0 

+ 

1 
V3 

-1 

8 
3V3 

0 

0 

1 

0 

— 

+    1 
V3 

-t 

8 
3\/3 

0 

1 

0 

0 

+ 

+  CO 

+  00 

+  QO 

+ 

cavity  changes  from  downward  to  upward.  The  following  numerical 
table  will  help  to  show  the  mode  of  variation  of  the  ordinate  and  of  the 
slope,  and  the  direction  of  bending. 

As  X  increases  from  —  oo  to 

V3 
the  bending  is  positive,  and  the  slope 
continually  increases  from  -co  through 

zero  to  a  maximum  value  — — ,  which 

3V3 
is  the  slope  of  the  stationary  tangent 

drawn  at  the  point  ( ,    -  )  • 

^  V      V3     9/ 

As    X   continues  to  increase  from 

to  H 3 ,  the  bending  is  nega- 

V3  V3 

tive,   and   the   slope    decreases  from 

Q 

H r:  through  zero  to  a  minimum 

3V3 

value  ^^,  which  is  the  slope  of  the  stationary  tangent  at  f  +  -— ,   -  J- 

Finally,  as  x  increases  from  +  —  to  +  oo,  the  bending  is  positive 

V3 
and  the  slope  increases  from  the  value 

Q 

through  zero  to  +  co. 

3V3 

The  values  a:  =  -  1,  0,  +1,  at 
which  the  slope  passes  through  zero, 
correspond  to  turning  values  of  the 
ordinate. 

Ex.  2.  Examine  for  inflexions  the  fiq.  39. 

curve        ar  +  4  =  (y  -  2)\ 

In  this  case 

2^  =  2+(x  +  4)^, 


Fig.  40. 


^  is  positive,  and  when  a:>-  4,  ^  is  negative. 


dy 
Hence,  at  the  point  (—  4,  2),  ~ 

d^v 
and  -r4  are  infinite.     When  j;<—  4, 
dx^ 

dh, 


156 


DIFFERENTIAL   CALCULUS 


[Ch.  XIII. 


Thus  there  is  a  point  of  inflexion  at  (  —  4,  2),  at  which  the  slope  is 
infinite,  and  the  bending  changes  from  the  positive  to  the  negative 

direction. 


Ex.  3.   Consider  the  curve  y=x*. 
dx  dx^ 


Fig.  41. 


At  (0,  0),  ^   is  zero,  but    the 

curve  has  no  inflexion,  for  — ^  never 

dx^ 
changes  sign  (Fig.  41). 


98.  Analytical  derivation  of  the  test  for  the  direction  of 
bending.  Let  the  equation  of  a  curve  be  i/  =/(2;),  and  let 
P,  ^j,  ^j),  be  a  point  upon  it.  Then  the  equation  of  the 
tangent  at  P  is 

Suppose  that  when  x  changes  from  x^  to  x^  +  ^,  the  ordi- 
nate of  the  tangent  change  from 
^j  to  y',  and  that  of  the  curve 
from  2/i  to  y'' ;  then  it  is  pro- 
posed to  determine  the  sign  of 
the  difference  of  ordinates  y  — «/' 
corresponding  to  the  same  ab- 
scissa x^  +  h. 

By  Taylor's  theorem, 

and  from  the  above  equation  of  the  tangent, 

Hence  /  =  yi  +  ¥'(^i)  =  /(^i)+  ¥'(^i)' 

and  it  follows  that 

y"-y  =  f/"(^i)+-- 


Fig.  42. 


P 


97-99.] 


DIRECTION  OF  BENDING 


167 


When  h  is  made  sufficiently  small, /''(a:j)+  •••  will  have 
the  same  sign  as  /'^(^i);  but  the  factor  h^  is  always  positive, 
hence  when  f(x-^  is  positive,  y"  —  y'  is  positive,  and  thus 
the  curve  is  above  the  tangent  at  both  sides  of  the  point  of 
contact,  that  is,  the  concavity  is  upward.  Similarly,  when 
f"{x{)  is  negative,  the  concavity  is  downward. 

This  agrees  with  the  former  result. 

99.  Concavity  and  convexity  towards  the  axis.     A  curve 

is  said  to  be  convex  or  concave  toward  a  line,  in  the  vicinity 
of  a  given  point  on  the  curve,  according  as  the  tangent  at 
the  point  does  or  does  not  lie  between  the  curve  and  the 
line,  for  a  finite  distance  on  each  side  of  the  point  of  contact. 


Fig.  43  a.  Fig.  43  6. 

First,  let  the  curve  be  convex  toward  the  a^axis,  as  in  the 
left-hand  figure.  Then  if  y  is  positive,  the  bending  is  positive 
and  —^■  is  positive ;  but  if  y  is  negative,  the  bending  is  neg- 
ative  and  — -  is  negative.  Hence  in  either  case  the  product 
y-zTT,  is  positive. 

Next,  let  the  curve  be  concave  toward  the  a:-axis,  as  in 
the  right-hand  figure.  Then  if  y  is  positive,  the  bending  is 
negative  and  -^  is  negative ;  but  if  y  is  negative,  the  bend- 
ing  is  positive  and  -j^  is  positive.  Thus  in  either  case  the 
product  y-^  is  negative.     Hence: 


158  DIFFERENTIAL   CALCULUS  [Ch.  XIII.  99. 

In  the  vicinity  of  a  given  point  (x^  y)  the  curve  is  convex  or 

concave  to  the  x-axis,  according  as  the  product  y  — ^  is  positive 

. .  dor 

or  negative. 

EXERCISES  ON  CHAPTER  XIII 

1.  Examine  the  curve  ?/  =  2  —  3(a;  —  2)5  for  points  of  inflexion. 

2.  Show  that  the  curve  a^y  =  x(cfi  —  a:^)  has  a  point  of  inflexion  at 
the  origin. 

3.  Find  the  points  of  inflexion  on  the  curve  y  = ; — • 

TO 

4.  In  the  curve  ay  =  i!^,  prove  that  the  origin  is  a  point  of  inflexion 
if  m  and  n  are  positive  odd  integers. 

5.  Show  that  the  curve  y  =csin  -  has  an  infinite  number  of  points  of 
inflexion  lying  on  a  straight  line. 

6.  Show  that  the  curve  y{x^  ■\-  0."^)  =  x  has  three  points  of  inflexion 
lying  on  a  straight  line ;  find  the  equation  of  the  line. 

7.  If  3^2  =f(x)  be  the  equation  of  a  curve,  prove  that  the  abscissas  of 
its  points  of  inflexion  satisfy  the  equation 

[f'(x)y  =  2f(x)^f"(xy 

8.  Draw  the  part  of  the  curve  a^y  =  ^-  ax^  +  2  a*  near  its  point  of 

3 

inflexion,  and  find  the  equation  of  the  stationary  tangent. 


CHAPTER   XIV 

CONTACT  AND  CURVATURE 

100.  Order  of  contact.  The  points  of  intersection  of  the 
two  curves 

are  found  by  making  the  two  equations  simultaneous ;  that 
is,  by  finding  those  values  of  x  for  which 

Suppose  ic  =  «  is  one  value  that  satisfies  this  equation. 
Then  the  point  x  —  a^  q^  =  (j)  (^a}  =  yjr  (^a)  is  common  to  the 
curves. 

If,  moreover,  the  two  curves  have  the  same  tangent  a^ 
this  point,  they  are  said  to  touch  each  other,  or  to  have 
contact  of  the  first  order  with  each  other.     The  values  of  y 

and  of  -^  are  thus  the  same  for  both  curves  at  the  point  in 
question,  which  requires  that 

<^  (a)  =  i/r  (a),      • 

If,  in  addition,  the  value  of  -t4  be  the  same  for  each 
curve  at  the  point,  then 

<^"(«)  =  f"(a), 
and  the  curves  are  said  to  have  a  contact  of  the  second 
order  with  each  other. 

If  <^(a)  =  i/r(a),  and  all  the  derivatives  up  to  the  nth. 
order  inclusive  be  equal  to  each  other,  the  curves  are  said 

159 


160  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

to  have  contact  of  the  nth  order.  This  is  seen  to  require 
n-^l  conditions.  Hence  if  the  equation  of  the  curve 
y  =  <i>(x)  be  given,  and  if  the  equation  of  a  second  curve 
be  written  in  the  form  y  =  '>^(x)',  in  which  "^^Qc)  proceeds 
in  powers  of  x  with  undetermined  coefficients,  then  n-\-l 
of  these  coefficients  could  be  determined  by  requiring  the 
second  curve  to  have  contact  of  the  nth.  order  with  the 
given  curve  at  a  given  point. 

101.  Number  of  conditions  implied  by  contact.  A  straight 
line  has  two  arbitrary  constants,  which  can  be  determined 
by  two  conditions  ;  accordingly  a  straight  line  can  be  drawn 
which  touches  a  given  curve  at  any  specified  point.  For  if 
the  equation  of  a  line  be  written  i/=mx-\-b,  then 

hence,  through  any  arbitrary  point  x  =  a  on  a  given  curve 

«/  =  (^(ic),  a  line  can  be  drawn  which  has  contact  of  the  first 

order  with  the  curve,  but  which  has  not  in  general  contact 

of  the  second  order ;  for  the  two  conditions  for  first-order 

contact  are 

ma  -\-b  =  <l>  (a), 

m  =  <^'(a), 

which  are  just  sufficient  to  determine  m  and  b. 

In  general  no  line  can  be  drawn  having  contact  of  an 
order  higher  than  the  first  with  a  given  curve ;  but  there 
are  certain  points  at  which  this  can  be  done.  For  example, 
the  additional  condition  for  second-order  contact  is  0  =  <f>"(a)^ 
which  is  satisfied  when  the  point  a:  =  a  is  a  point  of  inflexion 
on  the  given  curve  y  =  <t>(x)'  Thus  the  tangent  at  a  point 
of  inflexion  on  a  curve  has  contact  of  the  second  order 
with  the  curve. 


100-102.]  CONTACT  AND  CURVATURE  161 

The  equation  of  a  circle  has  three  independent  constants. 
It  is  therefore  possible  to  determine  a  circle  having  contact 
of  the  second  order  with  a  given  curve  at  any  assigned 
point. 

The  equation  of  a  parabola  has  four  constants,  hence  a 
parabola  can  be  found  which  has  contact  of  the  third  order 
with  the  given  curve  at  any  point. 

The  general  equation  of  a  central  conic  has  five  inde- 
pendent constants,  hence  a  conic  can  be  found  which  has 
contact  of  the  fourth  order  with  a  given  curve  at  any 
specified  point. 

As  in  the  case  of  the  tangent  line,  special  points  may  be 
found  for  which  these  curves  have  contact  of  higher  order. 

102.  Contact  of  odd  and  of  even  order. 

Theorem.  At  a  point  where  two  curves  have  contact  of 
an  odd  order  they  do  not  cross  each  other ;  but  they  do 
cross  where  they  have  contact  of  an  even  order. 

For,  let  the  curves  y  =  (f>(x)^  yz='\^(x)  have  contact  of 
the  nth  order  at  the  point  whose  abscissa  is  a ;  and  let  ^j, 
^2  be  the  ordinates  of  these  curves  at  the  point  whose 
abscissa  is  a  +  A.     Then 

and  by  Taylor's  theorem 

2,1  =  .^(a)  +  f  (a)  .  A +^!^  .  ^2 +... 


*^-^"+7;Sm^-«-^-- 


-^•^"-(^•^-'(«>-- 


162  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

Since  by  hypothesis  the  two  curves  have  contact  of  the 
wth  order  at  the  point  whose  abscissa  is  a,  hence 

and       y^-y^=  (^  +  1)!^'^°^'^"^  +  -  "  ^'^''^"'^  — '^  \ 

but  this  expression,  when  h  is  sufficiently  diminished,  has 
the  same  sign  as 

Hence,  if  n  be  odd,  y^  —  y^  does  not  change  sign  when  h  is 
changed  into  —  A,  and  thus  the  two  curves  do  not  cross  each 
other  at  the  common  point.  On  the  other  hand,  if  ti  be 
even,  y-^  —  y^  changes  sign  with  h ;  and  therefore  when  the 
contact  is  of  even  order  the  curves  cross  each  other  at  their 
common  point. 

For  example,  the  tangent  line  usually  lies  entirely  on  one 
side  of  the  curve,  but  at  a  point  of  inflexion  the  tangent 
crosses  the  curve. 

Again,  the  circle  of  second-order  contact  crosses  the 
curve  except  at  the  special  points  noted  later,  in  which 
the  circle  has  contact  of  the  third  order. 


EXERCISES 

1.  Find  the  order  of  contact  of  the  curves        r/A' 

4iy  =  x^  and  y  =  x  —  1.  f         jy*^ 

2.  Find  the  order  of  contact  of  the  curves        /»      'vtVr  v'l^^t^^ 

x^f  and  x  +  y  +  1  =  ^,fifX^  fi^^^ 

3.  Find  the  order  of  contact  of  the  curves 

43^  =  ^:2-4   and  x^-2y  =  S-i/^ 

4.  Determine  the  parabola  having  its  axis  parallel  to  the  y-axis, 
which  has  the  closest  possible  contact  with  the  curve  ah/  =  x^  at  the 
point  (a,  a). 


102-104.]  CONTACT  AND   CURVATUEE  163 

5.  Determine  a  straight  line  which  has  contact  of  the  second  order 
with  the  curve 

y  =  a:8-3a;2-  9a;  +  9. 

j6.   Find  the  order  of  contact  of 

y  =  log  (x  -  1)   and  x^  -  Qx -{■  2y -{■  S  =  0 
at  the  point  (2,  0). 

7.  What  must  be  the  value  of  a  in  order  that  the  curves 
y  =:  X  +  1  +  a(^x  —  ly  and  xy  =  'dx  —  1 
may  h^pvB  contact  of  the  second  order? 

103.  Circle  of  curvature.  The  circle  that  has  contact  of 
the  closest  order  with  a  given  curve  at  a  specified  point  is 
called  the  osculating  circle  or  circle  of  curvature  of  the 
curve  at  the  given  point.  The  radius  of  this  circle  is  called 
the  radius  of  curvature,  and  its  center  is  called  the  center 
of  curvature  at  the  assigned  point. 

104.  Length  of  radius  of  curvature;  coordinates  of  center 
of  curvature.     Let  the  equation  of  a  circle  be 

(X-«)2+(r-;S)2  =  i2^  (1) 

in  which  R  is  the  radius,  and  ct,  /3  are  the  coordinates  of  the 
center,  the  current  coordinates  being  denoted  by  X,  T  to 
distinguish  them  from  the  coordinates  of  a  point  on  the 
given  curve.  — 

It  is  required  to  determine  i2,  «,  yS,  so  that  this  circle 
may  have  contact  of  the  second  order  with  the  given  curve 
at  the  point  (a;,  y). 

From  (1),  by  successive  differentiation,  it  follows  that 


(2) 


164  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

If  the  circle  (1)  has  contact  of  the  second  order  at  the 
point  (a;,  ^)  with  the  given  curve,  then  when  X=x  it  is 
necessary  that 

Y=y.  1 

dX      dx     dX^     d^  J 
Substituting  these  expressions  in  (2), 


(a:-«)  +  (2/-^)g  =  0, 


(4) 


whence 

^  \dx)  d:^        \dx)  J  .. 

and  finally,  by  substitution  in  (1), 


v<m 


t^a;2 


105.  Direction  of  radius  of  curvature.     Since,  at  any  point 

P  on  the  given  curve,  tlie  value  of  -^  is  the  same  for  the 

dx 

curve  and  the  osculating  circle  for  that  point,  it  follows  that 

they  have  the  same  tangent  and  normal  at  P,  and  hence 

that   the  radius  of   curvature   coincides  with   the   normal. 

Again,  since  the  value  of  -^  is  the  same  for  both  curves  at 

dar 

P,  it  follows  from  Art.  97,  that  they  liave  tlie  same  direction 


104-105.] 


CONTACT  AND  CURVATURE 


165 


of  bending  at  that  point,  and  hence  that  the  center  of 
curvature  lies  on  the  concave  side  of  the  given  curve 
(Fig.  44). 

It  follows  from  this  fact  and  Art.  102  that  the  osculating 
circle  is  the  limiting  position  of  a  circle  passing  through 
three  points  on  the  curve  when  these  points  move  into 
coincidence. 

The  radius  of  curvature  is  usually  regarded  as  positive  or 
negative  according  as  the  bending  of  the  curve  is  positive 


Fig.  44. 


Fig.  45. 


or  negative  (Art*  97),  that  is,  according  as  the  value  of 


dx^ 


is  positive  or  negative ;  hence,  in  the  expression  for  H,  the 
radical  in  the  numerator  is  always  to  be  given  the  positive 
sign.  The  sign  of  It  changes  as  the  point  P  passes  through 
a  point  of  inflexion  on  the  given  curve  (Fig.  45).  It  is 
evident  from  the  figure  that  in  this  case  M  passes  through 
an  infinite  value  ;  for  the  circle  through  the  points  iV,  P,  Q 
approaches  coincidence  with  the  inflexional  tangent  when  N 
and  Q  approach  coincidence  with  P,  and  the  center  of  this 
circle  at  the  same  time  passes  to  infinity. 


166  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

106.  Total  curvature  of  a  given  arc;  average  curvature. 
The  total  curvature  of  an  arc  PQ  (Fig.  46)  in  which  the 

bending  is  continuous  and  in  one  direc- 
tion, is  the  angle  through  which  the 
tangent  swings  as  the  point  of  contact 
moves  from  the  initial  point  P  to  the 
terminal  point  Q\   or,  in  other  words, 
it  is  the  angle  between  the  tangents  at 
P  and  §,  measured  from  the  former  to 
the  latter.     Thus  the  total  curvature  of  a  given  arc  is  posi- 
tive or  negative  according  as  the  bending  is  in  the  positive 
or  negative  direction  of  rotation. 

The  total  curvature  of  an  arc  divided  by  the  length  of  the 
arc  is  called  the  average  curvature  of  the  arc.  Thus,  if  the 
length  of  the  arc  PQ  be  As  centimeters,  and  if  its  total 

curvature  be  A<^  radians,  then  its  average  curvature  is  — ^ 
radians  per  centimeter. 

107.  Measure  of  curvature  at  a  given  point.  The  mea^sure 
of  the  curvature  of  a  given  curve  at  a  given  point  P  is  the 
limit  which  the  average  curvature  of  the  arc  PQ  approaches 
when  the  point  Q  approaches  coincidence  with  P, 

Since  the  average  curvature  of  the  arc  P^  is  — ^,  the 
measure  of  the  curvature  at  the  point  P  is 

lim    A^ d^ 

'^  =  A.  =  o;^-^,' 

and  may  be  regarded  as  the  rate  of  deflection  of  the  arc  from 
the  tangent  estimated  per  unit  of  length;  or  again,  as  the 
ratio  of  the  angular  velocity  of  the  tangent  to  the  linear 
velocity  of  the  point  of  contact. 


106-108.] 

CONTACT  AND   CURVATURE                          167 

To  expri 

ess  K  in  terms  of  x^  y^  and  their  derivatives,  observe 

that 

ax 

Whence 

dx 

and 

as      d8\         dx) 

.      -£('•»-'£) -f 

da^            1 

\dxj      dx 


therefore  «  =  ^  = — -.  [Art.  83 

108.  Curvature  of  osculating  circle.  A  curve  and  its  oscu- 
lating circle  at  P  have  the  same  measure  of  curvature  at 
that  paint. 

For,  let  K^  k'  be  their  respective  measures  of  curvature  at 
the  point  of  contact  (a:,  y).     Then  from  Art.  107, 

dx^ 


K  = 


l'-(l)T 

But  this  is  the  reciprocal  of  the  expression  for  the  radius 
of  curvature  (Eq.  (6),  p.  164) ;  hence 


1 


168  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

That  is:  the  measure  of  curvature  k  at  a  point  P  is  the 
reciprocal  of  the  radius  of  curvature  R  for  that  point.  Since 
a  curve  and  its  osculating  circle  have  the  same  radius  at 
their  point  of  contact,  it  follows  from  this  result  that  the 
measure  of  curvature  is  also  the  same  for  both. 

It  is  on  account  of  this  property  that  the  osculating  circle 
is  called  the  circle  of  curvature.  This  is  sometimes  used  as 
the  defining  property  of  the  circle  of  curvature.  The  radius 
of  curvature  at  P  would  then  be  defined  as  the  radius  of  the 
circle  whose  measure  of  curvature  is  the  same  as  that  of  the 
given  curve  at  the  point  P.  Its  value,  as  found  from  Art. 
106  and  Art.  107,  accords  with  that  given  in  Art.  104. 

EXERCISES 

1.  Find  the  radius  of  curvature  of  the  curve  y^  =  4cax  at  the  origin. 

2.  Find  the  radius  of  curvature  of  the  curve  y^+3fi-^a(x'^-\-y^)  =  a^y 
at  the  origin. 

3.  Find  the  radius  of  curvature  of  the  curve  a^y  =  hx^  +  car^y  at  the 
origin. 

Find  the  radius  of  curvature  for  each  of  the  following  curveer: 

4.  xy  =  m^.    Rectangular  hyperbola. 

X^        7/2 

^'  ^-|2=1-     Hyperbola. 

6  a^-^y  =  x**.     General  parabola. 

7.  y/x  ■\-y/y  =\/a.     Parabola. 

V  8.  x^  +  y^  =  al     Hypocycloid. 

9.  y2  =  ^  ^     .    Cisaoid. 
2a  —  X 


10.  y  =  ^  (€•  +  c"«).    Catenary. 


K^^^-^ 


r 


108-109. 


CONTACT  AND  CURVATURE 


169 


109.   Direct  derivation  of  the  expressions  for  k  and  M  in 
polar  coordinates.     Using  the  notation  of  Art.  81, 


</>  =  6>  +  ^/r, 


hence 


""  c^s       ds_ 

dd 


'*  O-S) 


dO 


(•-^) 


[^m 

But                        tan  -^  =  p  — ,     y^r  —  tan-^ 

dp 

je^ 

tlierefore,  by  differentiating  as  to  6  and  reducing, 

(dp\^         cPp 

d^    \deJ     Pd0^ 

which,  substituted  in  (1),  gives 

p'-Pw  +  \Te) 

(1) 


[Art.  87. 


^-©7 


Since  «:  =  — ,  it  follows  that 
R 


E  = 


['■-s: 


de»^\d0) 


K  = 


170  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 

When  M  ==  -  is  taken  as  dependent  variable,  the  expres- 
sion for  K  assumes  the  simpler  form 

Since  at  a  point  of  inflexion  k  vanishes  and  changes  sign, 
hence  the  condition  for  a  point  of  inflexion,  expressed  in 

in 

polar  coordinates,  is  that  u  4-  -j^  shall  vanish  and  change 
sign. 

EXERCISES 
Find  the  radius  of  curvature  for  each  of  the  following  curves : 

1.  p  =  a^.  3.  /o  =  2acos^-a.        5.  p^cos2e=o^, 

2.  p?  =  aaco82^.        4.  pcos2^^  =  a.  6.  p  =  2a(l-co8^. 

7.  p6  =  a. 

'\\J\^^^  E VOLUTES  AND  INVOLUTES 

oi  110.  Definition  of  an  evolute.     When  the  point  P  moves 

along  the  given  curve,  the  center  of  curvature  Q  describes 
another  curve  which  is  called  the  evolute  of  the  first. 

Let  f(x^  y)=  0  be  the  equation  of  the  given  curve.  Then 
the  equation  of  the  locus  described  by  the  point  C  is  found 
by  eliminating  x  and  y  from  the  three  equations 


dx 
X  —  a  = 


d^ 


1  + 


y-P  = 


<i]L 


109-110.] 


CONTACT  AND   CURVATURE 


171 


and  thus  obtaining  a  relation  between  a,  yS,  the  coordinates 
of  the  center  of  curvature. 

No  general  process  of  elimination  can  be  given;  the 
method  to  be  adopted  depends  upon  the  form  of  the  given 
equation  f(x^  y)  =  0. 

Ex.  1.  Find  the  evolute  of  the  parabola  y^  =  4px. 

Since  y  =  2p^x^,  -^=p^x~^,  T^  =  -  t^P^^"^* 

ax  dx^         2 

hence         x—  a  =  —p^x~^  (1  +px~^)  2p~ix^  =  —  2{x  +  p), 
and  y  -  )8  =  (1  +  px-^)  2p~^x^  =  2  (p~^x^  +  /> W) ; 

therefore  a  =  2p-\-3Xf  P  =  -  2p~^xk 


Fig.  47. 


The  result  of  eliminating  x  between  the  last  two  equations  is 

M^-^py=Kp^m 

t.e.,  4(a-2py  =  2^pl3^y 


172  DIFFEEENTTAL   CALCULUS  [Ch.  XIV. 

which  is  the  equation  of  the  evolute  of  the  parabola,  a,  /3  being  the 
current  coordinates. 

Ex.  2.   Find  the  evolute  of  the  ellipse 

Here  £+iL.^  =  0,     f^^_^, 

a^      b^     dx  dx         a'^y 


y-x 


dy 


d^y         fe2     ^     -dx      -by         b^xH      -6^2  2.    ;2  2N      -^* 


dx^~        a^  "         ~     -o   o\./     I        o.     I—     .s.9\"-a      /—       •>    a> 


-by      ,   b^xH      -h\  ....  2,      -6* 
a^y^V       a^y  j      ahj^^    ^  '      a^y^ 

whence 

^      ^  a%*  \  b^  ^  a'r\   M  ^        2)-'r 

Therefore  ^  B  =  ^—r^y^  (2) 

Similarly,  a  =  ^—J^x^-  (3) 

Eliminating  x,  y  between  (1),  (2),  (3),  the  equation  of  the  locus 
described  by  («,  p)  is 

(aa)t  +  (bjS)^  =  (a2  -  J^)!.  (Fig.  52) 

111.  Properties  of  the  evolute.  The  evolute  has  two  im- 
portant properties  that  will  now  be  established. 

I.  The  normal  to  the  curve  is  tangent  to  the  evolute.  The 
relations  connecting  the  coordinates  (a,  yS)  of  the  center  of 
curvature  with  the  coordinates  (a:,  ?/)  of  the  corresponding 
point  on  the  curve  are,  by  Art.  104, 

rr-«  +  (y-/3)g  =  0,  (1) 

By  differentiating  (1)  as  to  x,  consideriucf  «,  /Q,  y  as 
functions  of  rr. 


110-111.]  CONTACT  AND   CURVATURE  173 

Subtracting  (3)  from  (2), 

*!  +  ^^  =  0,  (4) 

ax      ax  ax 

whence  di^_dx^ 

da  dy 

But  -y-  is  the  slope  of  the  tangent  to  the  e volute  at  (a,  y8), 

dir 
and  —  ;t-  is  the  slope  of  the  normal  to  the  given  curve  at 

(a;,  y).  Hence  these  lines  have  the 
same  slope;  but  they  pass  through 
the  same  point  (a,  /8),  therefore  they 
are  coincident. 


II.     The     difference     "between    two 
radii  of  curvature  of  the  given  curve^  \  p     ^ 

which  touch  the  evolute  at  the  points 
Cy,    C^  (^Fig.  4^),  is  equal  to  the  arc  O^Q^   of  the   evolute. 

Since  B  is  the  distance  between  points  (x^  «/),  (a,  yS), 
hence 

(x-ay^+<j,-py^=ii?.  (6) 

When  the  point  (x^  y)  moves  along  the  given  curve,  the 
point  (a,  /3)  moves  along  the  evolute,  and  thus  a,  y8,  i2,  y 
are  all  functions  of  x. 

Differentiation  of  (5)  as  to  x  gives 

(.-«)(i-|)-^c.-.)(|-f)=«f;      (6) 

hence,  subtracting  (6)  from  (1), 


174  DIFFERENTIAL   CALCULUS  [Ch.  XIV. 


(8) 


Again,  from  (1)  and  (4), 

da 

dx 

•          = 

d^ 

dx 

= =' 

Hence,  each  of  these  fractions  is  equal  to 


w^ 


dg\^  da 

^"^        =±^,  (9) 


V(a;-«)2+(2/-/3)2  B 

in  which  a  is  the  arc  of  the  evolute.     (Compare  Aft.  64.) 

Next,  multiplying  numerator  and  denominator  of  the  first 
member  of  (8)  by  a;  —  a,  and  those  of  the  second  member  by 
«/  —  yS,  and  combining  new  numerators  and  denominators,  it 
follows  that  each  of  the  fractions  in  (8)  is  equal  to 


(^_a)2+(^_^)2     ' 

which  equals  — 

j.dR 

dx 
-^  by  (7)  and  (5). 

By  combining 

:  with  (9), 

, 

da_^dR^ 
dx      ■*"  dx 

that  is, 

ax 

Therefore 

a  ±R  =  constant, 

(10) 

wherein  <r  is  measured  from  a  fixed  point  A  on  the  evolute. 
Now,  let  (7^,  C2  be  the  centers  of  curvature  for  the  points 


111.] 


CONTACT  AND   CURVATURE 


175 


^1'  -^2  ^2  "~  ^2  ' 


Pj,  Pg  o^  t^^  given  curve ;  let  Pj  (7^ 

let  the  arcs  -4  6\,  ^  63  be  denoted  by  o-^,  o-g.     Then 

0-1  ±  i^i  =  0-2  ±  E^,  by  (10); 
that  is,  o-j  —  o-g  =  ±  (i?2  —  -^1)5 

hence,  arc  C-^ C^=  R^  —  H^. 

Thus,  in  Fig.  49, 

PjCj  +  CjCg  =  P2^2' 
P9^'3!+  ^9^a  =  Pa^ai  ^tc. 


and 


(11) 


2^2 


2^3 


Fig.  49. 


Hence,  if  a  thread  be  wrapped 
around  the  evolute,  and  then  be 
unwound,  the  free  end  of  it  can  be  ^^ 
made  to  trace  out  the  original  curve. 
From  this  property  the  locus  of  the 
centers  of  curvature  of  a  given  curve 
is  called  the  evolute  of  that  curve,  and  the  latter  is  called 
the  involute  of  the  former. 

When  the  string  is  unwound,  each  point  of  it  describes  a 
different  involute ;  hence,  any  curve  has  an  infinite  number 
of  involutes,  but  only  one  evolute. 

Any  two  of  these  involutes  intercept  a  constant  distance 
on  their  common  normal,  and  are  called  parallel  curves  on 
account  of  this  property. 

Ex.  Find  the  length  of  that  part  of  the  evolute  of  the  parabola  which 
lies  inside  the  curve. 

From  Fig.  47  the  required  length  is  twice  the  difference  between  the 
tangents  C3P3  and  PqC^,  both  of  which  are  normals  to  the  parabola. 

To  find  the  coordinates  of  the  point  P^,  write  the  equation  of  the  tan- 
gent to  the  evolute  at  C3,  and  find  the  other  point  at  which  it  intersects 
the  parabola. 

The  coordinates  of  Cg,  the  point  of  intersection  of  the  two  curves,  are 
(8  JO ;  4:j9  V2),  and  the  equation  of  the  tangent  at  Cg  is 


176  DIFFERENTIAL   CALCULUS  [Ch   XIV. 

This  tangent  intersects  the  parabola  at  the  point  (2p,  —  2V2jo), 
which  is  Pg. 

The  value  of  the  radius  of  curvature  is  -v^"^_^)  ,  hence  PqCq  =  2p, 

PgCg  =  6\/3jt?,  hence  the  arc  C^C^  is  2jt)(3V3-l),  and  the  required 
length  of  the  evolute  is  therefore  4:p  (3  V3  —  1). 

EXERCISES 

Find  the  coordinates  of  the  center  of  curvature  for  each  of  the  follow- 
ing curves : 

'^1.   x^-\-y^=a^.  3.   y^=a^x. 

2.    x=alog^±:^^iHj!_V52T:p.         4.   y  =  |(J+r-). 

Find  the  equations  of  the  evolutes  of  the  following  curves : 

/"    5.   xy=a^.  /  6.   a^y'^ -1^"^  =  - a%\  /7.   a:t  +  ?/l  =  ai 

8.  Show  that  the  curvature  of  an  ellipse  is  a  minimum  at  the  end 
of  the  minor  axis,  and  that  the  osculating  circle  at  this  point  has  con- 
tact of  the  third  order  with  the  curve. 


Fio.  fiO. 


This  circle  of  curvature  must  be  entirely  outside  the  ellipse  (Fig.  50). 
For,  consider  two  points  Pj,  P,,  one  on  each  side  of  -C,  the  end  of  the 


111.] 


CONTACT  AND   CURVATURE 


177 


minor  axis.  At  these  points  the  curvature  is  greater  than  at  J5,  hence 
these  points  must  be  farther  from  the  tangent  at  B  than  the  circle  of 
curvature,  which  has  everywhere  the  same  curvature  as  at  J5. 

9.  Similarly,  show  that  the  curvature  at  Ay  the  end  of  the  major 
axis,  is  a  maximum,  and  that  the  circle  of  curvature  at  A  lies  entirely 
within  the  ellipse  (Fig.  50). 

10.  Show  how  to  sketch  the  circle  of  curvature  for  points  between  A 
and  B.  The  circle  of  curvature  for  points  between  A  and  B  has  three 
coincident  points  in  common  with  the  ellipse  (Art.  104),  hence  the  circle 
crosses  the  curve  (Art.  102).  Let  K,  P,  L  be  three  points  on  the  arc, 
such  that  K  is  nearest  A  and  L  nearest  B.    The  center  of  curvature  for 


Fig.  51. 


P  lies  on  the  normal  to  P,  and  on  the  concave  side  of  the  curve.  The 
circle  crosses  at  P,  lying  outside  of  the  ellipse  at  K  (on  the  side  towards 
A^,  and  inside  the  ellipse  at  L  ;  for  the  bending  of  the  ellipse  increases 
from  5  to  P  and  from  P  to  K,  while  the  bending  (curvature)  of  the 
osculating  circle  remains  constant  (Fig.  61).  * 

11.  Two  centers  of  curvature  lie  on  every  normal.     Prove  geometfi-  *^ 
cally  that  the  normals  to  the  curve  are  tangents  to  the  evolute.^   , '  "^ 

12.  Show  that  the   entire  length  of   the  evolute  of  the  ellipse  is 

4(~ ).     [From  equation  (11)  above,  take  i^j,  P,  as  the  radii  of 

curvature  at  the  extremities  of  the  major  and  minor  axes.] 


178 


DIFFERENTIAL   CALCULUS 


[Ch.  XIV.  111. 


13.   If  E  be  the  center  of  curvature  at  the  vertex  A  (Fig.  52),  prove 

that  CE  =  ae%  in  which  e 
is  the  eccentricity  of  the 
ellipse ;  and  hence  that  CD, 
CA,  CF,  CE  form  a  geo- 
metric series  whose  com- 
mon ratio  is  e.  Show  also 
that  DA,  AF,  FE  form  a 
similar  series. 

14.  If  H  be  the  center  of 
curvature  at  B,  show  that 
the  point  //  is  without  or 
within  the  ellipse,  according 
as  a  >  or  <  bV2,  or  accord- 
ing as  e^  >  or  <  ^. 

15.  Show  by  inspection 
of  the  figure  that  four  real 
normals  can  be  drawn  to 
the  ellipse  from  any  point 
within  the  evolute. 


CHAPTER   XV 
SINGULAR  POINTS 

112.  Definition    of    a    singular    point.      If  the   equation 
f(x^  ^)  =  0  be  represented  by  a  curve,  the  derivative  -5^, 

CLX 

when  it  has  a  determinate  value,  measures  the  slope  of  the 
tangent  at  the  point  (a;,  ?/).  There  may  be  certain  points 
on  the  curve,  however,  at  which  the  expression  for  the 
derivative  assumes  an  illusory  or  indeterminate  form  ;  and, 
in  consequence,  the  slope  of  the  tangent  at  such  a  point  can- 
not be  directly  determined  by  the  method  of  Art.  10.  Such 
values  of  x^  y  are  called  singular  values^  and  the  corre- 
sponding points  on  the  curve  are  called  singular  points. 

113.  Determination  of  singular  points  of  algebraic  curves. 

When  the  equation  of  the  curve  is  rationalized  and  cleared 

of  fractions,  let  it  take  the  form  f(x^  ^)  =  0. 

This   gives,  by   differentiation  with   regard   to  a;,  as  in 

Art.  71,  ,        ,  , 

•  ^+^^  =  0 
dx      dy  dx        ' 

dl 

whence  ^=-^.  (1) 

dy 
In    order  that  -^  may  become  illusory,  it   is   therefore 

necessary  that  §^=  ^'    F=^-  (^) 

179 


180  DIFFERENTIAL   CALCULUS  [Ch.  XV. 

Thus  to  determine  whether  a  given   curve  f(x^  ^)  =  ^ 

df  df 

has  singular  points,  put  -^  and  -^  each  equal  to  zero  and 

solve  these  equations  for  x  and  i/. 

If  any  pair  of  values  of  x  and  ^,  so  found,  satisfy  the 
equation  /(a;,  ^)  =  0,  the  point  determined  by  them  is  a 
singular  point  on  the  curve. 

To  determine  the  appearance  of  the  curve  in  the  vicinity 
of  a  singular  point  (x^,  y^),  evaluate  the  indeterminate  form 

di/ _      dx  _0 
^"""^■"0' 

by  finding  the  limit  approached  continuously  by  the  slope 
of  the  tangent  when  x^x^^  y  =  yv 

Hence  dy^_dx\dxj 

dx         i_(^ 

dx\dy) 

0  ^     ^  dy 

dx^      dxdy  dx  r  .  .„  „ 

"-    ay       ^fdy  [Arts.  49, 72. 

dx  dy      dy^  dx 
at  the  point  (a^j,  y{). 

This  equation  cleared  of  fractions  gives,  to  determine  the 
slope  at  (xy,  ^i),  the  quadratic 

This  quadratic  equation  has  in  general  two  roots.  The 
only  exceptions  occur  when  simultaneously,  at  the  point  in 
question, 

Bt?     ^'    dxdy       '    dy^       '  ^^ 


113-114.]  SINGULAR  POINTS  181 

in  which  case  -:r-  is  still  indeterminate  in  form,  and  must  be 
ax 

evaluated  as  before.     The  result  of  the  next  evaluation  is  a 

cubic  in  -^,  which  gives  three  values  to  the  slope,  unless  all 

the  third  partial  derivatives  vanish  simultaneously  at  the 
singular  point. 

The  geometric  interpretation  of  the  two  roots  of  equation 
(3)  will  now  be  given,  and  similar  principles  will  apply 
when  the  quadratic  is  replaced  by  an  equation  of  higher 
degree. 

The  two  roots  of  (3)  are  real  and  distinct,  real  and  coin- 
cident, or  imaginary,  according  as 


(: 


dx  By)      dx^  5^2 


is  positive,  zero,  or  negative.  These  three  cases  will  be 
considered  separately. 

114.  Multiple  points.     First  let  H  be  positive.     Then  at 

df  df 

the  point  (x,  y)  for  which  ^  =  0,  ^  =  0,  there  are  two  values 

ijx         oy 

of  the  slope,  and  hence  two  distinct  singular  tangents.  It 
follows  from  this  that  the  curve  goes  through  the  point  in 
two  directions,  or,  in  other  words,  two  branches  of  the  curve 
cross  at  this  point.  Such  a  point  is  called  a  real  double 
point  of  the  curve,  or  simply  a  node.  The  conditions,  then, 
to  be  satisfied  at  a  node  (a^j,  y-^  are 

and  H(x-^,  y{)  >  0. 

Ex.   Examine  for  singular  points  the  curve 

3  x^  -  xy  -  2  y^  +  x^  -  8y^  =  0, 


182 


DIFFERENTIAL   CALCULUS 


[Ch.  XV. 


Here  |f  =  6a:  -  v  +  3x2,  ^=  -  a:  -  4  v  -  24  v^. 

dx  oy  ^  ^ 

The  values  x  =  0,    ?/  =  0  will  satisfy  these  three  equations,  hence 
(0,  0)  is  a  singular  point. 


Since 


1^=6  + 6a:  =  6  at  (0,0), 

bxby  * 

^  =  _4-48y=-4at  (0,0), 


FiQ.  53. 


hence  the  equation  determining  the  slope  is,  from  (3), 

-(ir-(i)-«-. 

of  which  the  roots  are  1  and  —  f .    It  follows  that  (0,  0)  is  a  double 
point  at  which  the  tangents  have  the  slopes  1,  —  |. 

115.  Cusps.  Next  let  5"=  0.  The  two  tangents  are  then 
coincident,  and  there  are  two  cases  to  consider.  If  the 
curve  recedes  from  the  tangent  in  both  directions  from  the 
point  of  tangency,  the  singular  point  is  called  a  tacnode. 
Two  distinct  branches  of  the  curve  touch  each  other  at 
this  point.     (See  Fig.  54.) 

If  both  branches  of  the  curve  recede  from  the  tangent  in 
only  one  direction  from  the  point  of  tangency,  the  point  is 
called  a  cusp. 


114-115.] 


SINGULAR  POINTS 


183 


Here  again  there  are  two  cases  to  be  distinguished.  If 
the  branches  recede  from  the  point  on  opposite  sides  of  the 
double  tangent,  the  cusp  is  said  to  be  of  the  first  kind ;  if 
they  recede  on  the  same  side,  it  is  called  a  cusp  of  the  second 
kind. 

The  method  of  investigation  will  be  illustrated  by  a  few 
examples. 

Ex.  1.  f(x,  y)  =  aY  -  «^^*  +  a:«  =  0. 

dx  dy 

The  point  (0,  0)  will  satisfy  /(x,  y)=  0,  ^  =  0,  ^  =  0 ;  hence  it  is  a 
singular  point.     Proceeding  to  the  second  derivatives, 

-  12  a%2  +  30  a:*  =  0  at  (0,  0), 

^"f  =0 

dxdy        * 

dy'' 

The  two  values  of  -r-  are  therefore  coincident,  and  each  equal  to  zero. 
dx 

From  the  form  of  the  equation,  the  curve  is  evidently  symmetrical  with 

regard  to  both  axes;  hence  the  point  (0,  0)  is  a  tacnode. 

No  part  of  the  curve  can  be  at  a  greater  distance  from  the  y-axis  than 

±  a,  at  which  points  -^  is  infinite.     The  maximum  value  of  y  corre- 
dx 

sponds   to  x  =  ±aV\,     Between  a;  =  0,  ar  =  aV|  there  is  a  point  of 

inflexion  (Fig.  54). 

Ex.2.  /(a:,y)=3^2-.x8=0; 

|f=-3:r^,   f  =2^. 
dx  dy 


ay 

dx^ 


Hence  the  point  (0,  0)  is  a  singu- 
lar point. 


Further,  If,: 

ay 

dxby 


6a:=0at(0,0); 

0-  ^-2 


Fig.  54. 


184 


DIFFERENTIAL   CALCULUS 


[Ch.  XV. 


Therefore  the  two  roots  of  the  quadratic  equation  defining  -^  are  both 

dx 

equal  to  zero.     So  far,  this  case  is  exactly  like  the  last  one,  but  here  no 

part  of  the  curve  lies  to  the  left  of  the  axis  y.     On  the  right  side,  the 

curve  is  symmetric  with  regard  to  the  a:-axis.     As  x  increases,  y  increases; 

there  are  no  maxima  nor  minima,  and  no  inflexions  (Fig.  55). 


Ex.3. 


/(x,  y)z=x^-  2ax'^y  -  axy^  +  aV  ^  q. 


The  point  (0,  0)  is  a  singular  point,  and  the  roots  of  the  quadratic  defining 


dx 


are  both  equal  to  zero. 

Let  a  be  positive.     Solving  the  equation  for  y, 


When  X  is  negative,  y  is  imaginary ;  when  a:  =  0,  y  =  0 ;  when  x  is 
positive,  but  less  than  a,  y  has  two  positive  values,  therefore  two  branches 


Pig.  66. 


Pro.  66. 


are  above  the  a:-axis.  When  ar  =  a,  one  branch  becomes  infinite,  having 
the  asymptote  x  =  a]  the  other  branch  has  the  ordinate  \ a.  The  origin 
is  therefore  a  cusp  of  the  second  kind  (Fig.  56). 

116.  Conjugate  points.  Lastly,  let  H  be  negative.  In 
this  case  there  are  no  real  tangents ;  hence  no  points  in  the 
immediate  vicinity  of  the  given  point  satisfy  the  equation  of 
the  curve. 

Such  an  isolated  point  is  called  a  conjugate  point. 


115-116.] 


SINGULAR  POINTS 


185 


Ex.    f(x,  y)  =  ay^  —  a;^  +  hx^  =  0. 
a  singular 


Here  (0,  0)   is  a  singular  point  of  the 
locus,  and 


dx 

both  roots  being  imaginary  if  a  and  b 
have  the  same  sign. 

To  show  the  form  of  the  curve,  solve 
the  given  equation  for  y. 


Then 


=±x4 


Fig.  57. 


and  hence,  if  a  and  b  are  positive,  there 

are  no  real  points  on  the  curve  between  x  =  0  and  x  =  b.     Thus  0  is  an 

isolated  point  (Fig.  57). 

These  are  the  only  singularities  that  algebraic  curves  can 
have,  although  complicated  combinations  of  them  may  ap- 
pear. In  each  of  the  foregoing  examples,  the  singular  point 
was  (0,  0) ;  but  for  any  other  point,  the  same  reasoning  will 
apply. 

Ex.  f(x,  7/)=  x^  -{-  S  y^  -  U  y^  -  4:x  +  17  y  -  S  =  0, 


^=2x-4      ^: 
dx  '    dy 


y2_2Qy  +  l\ 


At  the  point  (2,  1),  /(2,  1)=  0,  %.  =  0,  ¥  =  0;    hence  (2,  1)  is  a 
singular  point.  ^ 


dV   _ 


^'^^    S  =  ^'     ^y  =  '-^    W='''-''^    =-8  at  (2,1). 
Hence  --p-  —  ±\\  and  thus  the  equations  of  the  two  tangents  at  the 
node  (2,  1)  are  y  -  1  =  i(a:  -  2),  y  -  1  =  -  K^  -  2). 

When  H  is  negative,  the  singular  point  is  necessarily  a 
conjugate  point,  but  the  converse  is  not  always  true.  A 
singular  point  may  be  a  conjugate  point  when  11=0. 
[Compare  Ex.  4  below.] 


186  DIFFERENTIAL   CALCULUS  [Ch.  XV.  116. 

EXERCISES  ON  CHAPTER  XV 

Examine  each  of  the  following  curves  for  multiple  points  and  find  the 
equations  of  the  tangents  at  each  such  point : 

1.   a2x2  =  ftV  +  ^ V- 


2. 

^       2a-x 

3. 

xt  +  yl  =ai. 

4. 

^2(x2  _  a-2)  =  x\ 

5.  y  z=za  +  X  -\-  hx^  ±  cx'i. 

When  a  curve  has  two  parallel  asymptotes  it  is  said  to  have  a  node  at 
infinity  in  the  direction  of  the  parallel  asymptotes.     Apply  to'Ex.  6. 

6.  (x^-y^)2-^y^+y  =  0.      ■ 

7.  a;4  -  2  a?/8  _  3  a2^2  _  2  a2a.2  ^  (j4  =  0. 

8.  y^  =  x(x-\-ay. 

9.  a;8  -  3  axy  -\-y^  =  0. 

10.  y^  =  x*~{-x^. 

11.  Show  that  the  curve  y  =  x\ogx  has  a  terminating  point  at  the 
origin. 


CHAPTER  XVI 
ENVELOPES 
117.  Family  of  curves.     The  equation  of  a  curve, 

usually  involves,  besides  the  variables  x  and  ^,  certain  coeffi- 
cients that  serve  to  fix  the  size,  shape,  and  position  of  the 
curve.  The  coefficients  are  called  constants  with  reference 
to  the  variables  x  and  «/,  but  it  has  been  seen  in  previous 
chapters  that  they  may  take  different  values  in  different 
problems,  while  the  form  of  the  equation  is  preserved.  Let 
a  be  one  of  these  "constants."  Then  if  a  be  given  a  series 
of  numerical  values,  and  if  the  locus  of  the  equation,  corre- 
sponding to  each  special  value  of  a  be  traced,  a  series  of 
curves  is  obtained,  all  having  the  same  general  character, 
but  differing  somewhat  from  each  other  in  size,  shape,  or 
position.  A  system  of  curves  so  obtained  is  called  a  family 
of  curves. 

For  example,  if  A,  h  be  fixed,  and  p  be  arbitrary,  the  equa- 
tion Qy —  k')'^  =  2p(x— K)  represents  a  family  of  parabolas, 
each  curve  of  which  has  the  same  vertex  (7a,  A;),  and  the 
same  axis  y=h^  but  a  different  latus  rectum.  Again,  if  k 
be  the  arbitrary  constant,  this  equation  represents  a  family 
of  parabolas  having  parallel  axes,  the  same  latus  rectum,  and 
having  their  vertices  on  the  same  line  x  =  h. 

The  presence  of  an  arbitrary  constant  a  in  the  equation  of 
a  curve  is  indicated  in  functional  notation  by  writing  the 

187 


188  DIFFERENTIAL   CALCULUS  [Ch.  XVI. 

equation  in  the  form  /(a;,  y^  «)  =  0.  The  quantity  «,  which 
is  constant  for  the  same  curve  but  different  for  different 
curves,  is  called  the  parameter  of  the  family.  The  equa- 
tions of  two  neighboring  curves  are  then  written 

f(x,  y,  a)  =  0,  f{x,  y,  a  +  h')=  0, 

in  which  A  is  a  small  increment  of  a.  These  curves  can  be 
brought  as  near  to  coincidence  as  desired  by  diminishing  h. 

118.  Envelope  of  a  family  of  curves.  A  point  of  inter- 
section of  two  neighboring  curves  of  the  family  tends  toward 
a  limiting  position  as  the  curves  approach  coincidence.  The 
locus  of  such  limiting  points  of  intersection  is  called  the 
envelope  of  the  family. 

Let  f(x,y,a)=0,        /(x,  y,  a+ h)==0  (1) 

be  two  curves  of  the  family.  By  the  theorem  of  mean  value 
(Art.  45) 

f(x,  y,a-hh^  =  fCx,  y,  a)-\-h^Cx,  y,  a-^-OK),  (2) 

da 

which,  on  account  of  equation  (1),  reduces  to 

Hence,  it  follows  that  in  the  limit,  when  A  =  0, 

is  the  equation  of  a  curve  passing  through  the  limiting 
points  of  intersection  of  the  curve  /(a:,  ?/,  a)  =  0  with  its 
consecutive  curve.  This  determines  for  any  assigned  value 
of  a  a  definite  limiting  point  of  intersection  on  the  corre- 
sponding member  of   the  family.      The  locus  of   all  such 


117-119.]  -EN  VEL  OPES  189 

points  is  then  to  be  obtained  by  eliminating  the  parameter 
a  between  the  equations 

/(a;,  y,  «)=  0,  Z(a;,  «/,  «)=  0. 
da 

The  resulting  equation  is  of  the  form  F(x^  y)  =  0,  and 
represents  the  fixed  envelope  of  the  family. 

119.  The  envelope  touches  every  curve  of  the  family. 

I.  Geometrical  'proof.  Let  A,  B^  Q  be  three  consecutive 
curves  of  the  family  ;  let  A^  B  intersect  in  P  ;  B^  C  inter- 
sect in  Q.  When  P,  Q  approach  coincidence,  PQ  will  be 
the  direction  of  the  tangent  to  the  envelope  at  P ;  but  since 


P,  Q  are  two  points  on  B  that  approach  coincidence,  hence 
P(>  is  also  the  direction  of  the  tangent  to  B  at  P,  and 
accordingly  B  and  the  envelope  have  a  common  tangent  at 
P.     Similarly  for  every  curve  of  the  family. 

II.    More  rigorous  analytical  proof.     Let  — f(x^  y,  a)  =  0 

da 

be  solved  for  a,  in  the  form  a=  <t>(x,  «/).  Then  the  equation 
of  the  envelope  is 

Equating  the  total  rr-derivative  to  zero, 

dx     dy  dx     d<p\Sx      by  dx) 


190  DIFFERENTIAL   CALCULUS  [Ch.  XVI. 

but  -=^  =  —  =  0,    hence   the   slope   of    the   tangent   to   the 
d(p      da 

envelope  at  the  point  (a;,  y)  is  given  by 

dx      by  dx 

But  this  equation  defines  the  direction  of  the  tangent  to 
the  curve  f(x^  y^  a)  =  0  at  the  same  point,  and  therefore  a 
limiting  point  of  intersection  on  any  member  of  the  family 
is  a  point  of  contact  of  this  curve  with  the  envelope. 

Ex.   Find  the  envelope  of  the  family  of  lines 


obtained  by  varying  m. 
Differentiate  (1)  as  to  w, 


y  =  mx+|'  .  (1) 


0  =  --^-  (2) 


To  eliminate  m,  multiply  (2)  by  m  and  square;  square  (1)  and  sub- 
tract the  first  from  the  second.     The  envelope  is  found  to  be  the  parabola 

y^  =  4c  px. 

120.  Envelope  of  normals  of  a  given  curve.  The  evolute 
(Art.  110)  was  defined  as  the  locus  of  the  centers  of  curva- 
ture. The  center  of  curvature  was  shown  to  be  the  point  of 
intersection  of  consecutive  normals  (Art.  Ill),  whence  by 
Art.  118  the  envelope  of  the  normals  is  the  evolute. 

Ex.   Find  the  envelope  of  the  normals  to  the  parabola  y"^  =  4jt)a;. 
The  equation  of  the  normal  at  (Xj,  yy)  is 

or,  eliminating  x^  by  means  of  the  equation  y^  =  4/>Xj, 

i/-v,  =  ^-^  (1) 


119-121.] 


ENVELOPES 


191 


The  envelope  of  this  line,  when  y^  takes  all  values,  is  required 
Differentiating  as  to  y^ 

~  Sp^      2p 

Substituting  this  value  for  y^  in  (1),  the  result^ 
27  py^  =  i(x  -  2 py, 
is  the  equation  of  the  required  evolute.  *• 


121.  Two  parameters,  one  equation  of  condition.  In  many 
cases  a  family  of  curves  may  have  two  parameters  which  are 
connected  by  an  equation.  For  instance,  the  equation  of 
the  normal  to  a  given  curve  contains  two  parameters  x^,  y-^ 
which  are  connected  by  the  equation  of  the  curve.  In  such 
cases  one  parameter  may  be  eliminated  by  means  of  the  given 
relation,  and  the^ other  treated  as  before. 

When  the  elimination  is  difficult  to  perform,  both  equa- 
tions may  be  differentiated  as  to  one  of  the  parameters  a, 
regarding  the  other  parameter  yS  as  a  function  of  a.     This 

dB 
gives  four  equations  from  which  a,  y8  and  -^  may  be  elim- 

da 

inated,  the   resulting  equation  being   that   of   the   desired 

envelope. 

Ex.  1.   Find  the  envelope  of  the  line 

a      b 

the  sum  of  its  intercepts  remaining  constant. 
The  two  equations  are 

X      y      ^ 
-  +  I  =  1, 
a      0 

a+b  =  c. 


192 


DIFFERENTIAL   CALCULUS 


[Ch.  XVI. 


Differentiate  both  equations  as  to  a ; 

1  +  ^  =  0. 
da 


Eliminate 


da 


Then  —  =  ^^    which  reduces  to 

^      I      X      y_ 

a      b      a      b      \         .  , —    ,         , — 

-  =  J-  = 7  =  -;    whence   a  —  y/cx,   b  =  Vcy. 

a      b      a+b      c'_  ^ 

Therefore  Va;  +  Vy  =  Vc 

is  the  equation  of  the  desired  envelope.     [Compare  Ex.  p.  131.] 

Ex.  2.   Find  the  envelope  of  the  family  of  coaxial  ellipses  having  a 
constant 
Here 


121.]  ENVELOPES  193 

For  symmetry,  regard  a  and  b  as  functions  of  a  single  parameter  t. 
Then  ^da+^db  =  0, 

bda  •+  ac?6  =  0 ; 

hence  —-  =  ^  =  --, 

a=±a:V2,    b=±yV2, 
and  the  envelope  is  the  pair  of  rectangular  hyperbolas  xy  =±^ k^. 

Note.  A  family  of  curves  may  have  no  envelope ;  i.e.,  consecutive 
curves  may  not  intersect;  e.g.,  the  family  of  concentric  circles  x^  +  y^=r^, 
obtained  by  giving  r  all  possible  values. 

If  every  curve  of  a  family  has  a  node,  and  the  node  has 
different  positions  for  different  curves  of  the  family,  the 
envelope  will  be  composed  of  two  (or  more)  curves,  one  of 
which  is  the  locus  of  the  node. 

Ex.   Find  the  envelope  of  the  system 

/=  (y-\y  +  x^-x^  =  0, 

in  which  A  is  a  varying  parameter. 

Here  -^  =  —  2(y  —  A,)  =  0 ;  by  combining  with  /=  0  to  eliminate  X, 

we  obtain 

a;2  =  0,   X  -  1  =  0,   x  +  1  =  0. 

From  Art.  114  it  is  seen  that 

x  =  0,  y  =  X 

is  a  node  on  /;  moreover,  the  various  curves  of  the  family  are  obtained 
by  moving  any  one  of  them  parallel  to  the  y-axis.  The  lines  a:  —  1  =  0, 
a:  +  1  =  0  form  the  proper  envelope,  and  a:  =  0  is  the  locus  of  the  node. 


EXERCISES  ON  CHAPTER  XVI 

1.  Find  the  envelope  of  the  line  x  cos  a  +  y  sin  «  =  jo,  when  ce  is  a 
parameter. 

2.  A  straight  line  of  fixed  length  a  moves  with  its  extremities  in  two 
rectangular  axes.     Find  its  envelope. 


194 


DIFFERENTIAL   CALCULUS  [Ch.  XVI.  121. 


^        3.   Ellipses  are  described  with  common  centers  and  axes,  and  having 
the  sum  of  the  serai-axes  equal  to  c.     Find  their  envelope. 

\/      4.   Find  the  envelope  of  the  straight  lines  having  the  product  of  their 
intercepts  on  the  coordinate  axes  equal  to  k\ 

/     5.   Find  the  envelope  of  the  lines  y  —  ^  =  m(x  —  a)  +  rVl  +  m^,  m 
being  a  variable  parameter. 

6.  A  circle  moves  with  its  center  on  a  parabola  whose  equation  is 
y2  =  4  axj  and  passes  through  the  vertex  of  the  parabola.  Find  its 
envelope. 

7.  Find  the  envelope  of  a  perpendicular  to  the  normal  to  the  parabola 
y2  =  4  ax,  drawn  through  the  intersection  of  the  normal  with  the  x-axis. 

8.  Show  that  the  curves  defined  by  the  equations 

^  +  ^=1,   a  +  p  =  c, 
X      y  "^ 

in  which  a  and  j8  are  parameters,  all  pass  through  four  fixed  points ;  find 
them. 

9.  In  the  « nodal  family ''  {y  -  2ay={x  -  aY  +  Sx^  -  y\  show  that 
the  usual  process  gives  for  envelope  a  composite  locus,  made  up  of  the 
"node-locus  "  (a  line)  and  the  envelope  proper  (an  ellipse). 


INTEGRAL   CALCULUS 

CHAPTER  I 
GENERAL  PRINCIPLES  OF  INTEGRATION 

122.  The  fundamental  problem.  The  fundamental  prob- 
lem of  the  Differential  Calculus,  as  explained  in  the  preced- 
ing pages,  is  this : 

Given  a  function  f  (^x),  of  an  independent  variable  x,  to 
determine  its  derivative  f'(x). 

It  is  now  proposed  to  consider  the  inverse  problem,  viz.  : 

Given  any  function  f'(x),  to  determine  the  function  fix) 
having  f  {x)  for  its  derivative. 

The  study  of  this  inverse  problem  is  one  of  the  objects 
of  the  Integral  Calculus. 

The  given  function  f'(x)  is  called  the  integrand^  the 
function  f(x)  which  is  to  be  found  is  called  the  integral^  and 
the  process  gone  through  in  order  to  obtain  the  unknown 
function  f(x)  is  called  integration. 

The  operation  and  result  of  differentiation  are  symbolized 
by  the  formula  , 

£/w=/'(^),  (1) 

or,  written  in  the  notation  of  differentials, 

dfix-)=f'ix)dx.  (2) 

195 


196  INTEGRAL   CALCULUS  [Ch.  I. 

The  operation  of  integration  is  indicated  by  prefixing  the 
symbol  j  to  the  function,  or  differential,  whose  integral  it 
is  required  to  find.  Accordingly,  the  formula  of  integration 
is  written  thus : 

Following  long  established  usage,  the  differential,  rather 
than  the  derivative,  of  the  unknown  function  f(x)  is  written 
under  the  sign  of  integration.  One  of  the  advantages  of  so 
doing  is  that  the  variable,  with  respect  to  which  the  inte- 
gration is  performed,  is  explicitly  mentioned.  This  is,  of 
course,  not  necessary  when  only  one  variable  is  involved, 
but  is  essential  when  several  variables  enter  into  the  inte- 
grand, or  a  change  of  variable  is  made  during  the  process 
of  integration. 

123.  Integration  by  inspection.  The  most  obvious  aid  to 
the  problem  of  integration  is  a  knowledge  of  the  rules  and 
results  of  differentiation.  It  frequently  happens  that  the 
required  function  f{x)  can  be  determined  at  once  by  recol- 
lecting the  result  obtained  in  some  previous  differentiation. 

For  example,  suppose  it  to  be  required  to  find 


/ 


cos  X  dx. 


It  will  be  recalled  that  cos  x  dx  is  the  differential  of  sin  x^ 
and  thus  the  answer  to  the  proposed  integration  is  directly 
obtained.     That  is, 

cos  xdx  ss  sin  x. 


f' 


Again,  suppose  it  is  required  to  integrate 


j  x"dx^ 


122-123.]      GENERAL    PRINCIPLES   OF  INTEGRATION  197 

where  n  is  any  constant  (except  —  1).  This  problem  imme- 
diately suggests  the  formula  for  differentiating  a  variable 
affected  by  a  constant  exponent  [(6),  p.  49].  When  this 
formula  is  written 

or,  what  is  the  same  thing, 


it  becomes  obvious  that 


/= 


x^dx  = 


n  +  1 

An  exception  to  this  result  occurs  when  n  has  the  value 
—  1.     For  in  that  case  it  is  apparent  from  (8),  p.  50,  that 

'dx 


ix'^dx—  \  —  =  log  X. 


The  method  indicated  in  the  above  illustration  may  be 
designated  as  the  method  of  integration  hy  inspection.  This 
is  in  fact  the  only  method  of  practical  service  available. 
The  object  of  the  various  devices  suggested  in  the  subse- 
quent pages  is  to  transform  the  given  integrand,  or  to 
separate  it  into  simpler  elements  in  such  a  way  that  the 
method  of  inspection  can  be  applied.* 

*  When  all  has  been  done  that  can  be  accomplished  in  this  direction,  it 
will  be  found  that  a  large  portion  of  the  field  is  yet  unexplored  and  unknown, 
and  that  many  functions  exist  whose  integrals  cannot  be  found.  By  .this  we 
mean  that  such  integrals  cannot  be  expressed  in  terms  of  functions  already 
known.  To  illustrate,  let  it  be  imagined  that  the  integral  calculus  had  been 
discovered  before  the  logarithm  function  was  known.     It  would  then  have 

I*  (It 
been  impossible  to  express  the  integral    \  —  in  terms  of  known  functions. 

This  integral  might  in  consequence  have  led  to  the  discovery  of  the  function 
log  X.  An  exactly  analogous  thing,  in  fact,  has  happened  in  the  attempt  to 
integrate  other  expressions,  and  many  important  and  hitherto  unknown  func- 
tions have  been  discovered  in  this  way  which  have  greatly  enriched  the  entire 
field  of  mathematics. 


198  INTEGRAL   CALCULUS  [Ch.  I. 

124.  The  fundamental  formulas  of  integration.  When  the 
formulas  of  differentiation  (l)-(26),  pp.  49-50,  are  borne  in 
mind,  the  method  of  inspection  referred  to  in  the  preceding 
article  leads  at  once  to  the  following  fundamental  integrals. 
Upon  these  sooner  or  later  every  integration  must  be  made 
to  depend. 

1^ 


I.    \u^du  =  ^^^^* 
n  +  1 


II.    f**  =  log«. 

III.    frt''efw  =  -^^. 
J  log  a 

lY.  (e^du  =  e^. 

T,  f  cos  u  du  =  sin  u, 

VI.  \^inudu  =  -eosu* 

Til.  f sec^  udu  =  tan  u, 

VIII.  f cosec'-^  udu  =  -  cot  u, 

IX.  f sec  u  tan  w  <fw  =  sec  u, 

X.  (  cosec  u  cot  u  du  =  —  cosec  u, 

XI.  r_^^_  =  sin-i  w,  or  -cos-^i*. 

XII.    f-^^^  =  tan-it*,  or  -cot-^w. 

XIII.  f — ^^       =  sec-'  u,  or  -cosec-^M, 
•^  u^u^  i- 1 

XIV.  J 


^^       =  rers-i  u. 


V  2  u  -  u« 


l:f 


:  /. 


L 


124-125.]      GENERAL  PRINCIPLES  OF  INTEGRATION  199 

125.  Certain  general  principles.  In  applying  the  above 
formulas  of  integration  certain  principles  which  follow  from 
the  rules  of  differentiation  should  be  borne  in  mind. 

(a)  The  integral  of  the  sum  of  a  finite  number  of  functions 
is  equal  to  the  sum  of  the  integrals  of  each  function  taken 
separately. 

This  follows  from  Art.  16. 

For  example, 

j__dx=^^xdx-}-  =  --logx. 

(5)  A  constant  factor  may  he  removed  from  one  side  of  the 
sign  of  integration  to  the  other. 

For,  since 

d(G  *  u)=  c  '  du, 
it  follows  that 

j  cdu  =  c  \du  =  cu. 

To  illustrate,  let  it  be  required  to  integrate 


f' 


Sx^dx. 


The  numerical  factor  5  is  first  placed  outside  the  sign  of 
integration,  after  which  formula  I  is  applied.     Accordingly, 


f5x^dx=5fx^dx  =  5'^' 


a 

Again,  suppose  the  integral 


J  x^4-l 


_  dx 

is  to  be  found.  It  is  readily  noticed  that  except  for  the 
constant  factor  2  the  numerator  of  the  integrand  is  the  exact 
derivative  of   the   denominator,  and   formula  II  would   be 


200  INTEGRAL   CALCULUS  [Ch.  I. 

applicable.  All  that  is  required,  then,  in  order  to  reduce 
the  given  integral  to  a  known  form,  is  to  multiply  inside  the 
sign  of  integration  by  2  and  outside  by  J.     This  gives 

r  xdx     .  r2xdx    1  rdOt^  +  r)    .,    /^  ,  in 

In  this  connection  it  must  not  be  forgotten  that  an  expres- 
sion containing  the  variable  of  integration  cannot  be  removed 
from  one  side  of  the  sign  of  integration  to  the  other. 

(c)  An  arbitrary  constant  may  be  added  to  the  result  of 
integration. 

For,  the  derivative  of  a  constant  is  zero,  and  hence 

du  =  d(u  +  c), 
from  which  follows 

I  du=:  i  d(u  -{-c^=u-\-o. 

This  constant  is  called  the  constant  of  integration. 

It  will  be  seen  from  this  that  the  result  of  integration  is 
not  unique,  but  that  any  number  of  functions  (differing  from 
each  other,  however,  only  by  an  additive  constant)  can  be 
found  which  have  the  same  given  expression  for  derivative. 
[Compare  Art.  16,  Cor.  2.] 

Thus,  any  one  of  the  functions  rr^  —  1,  rc^  +  1,  a^  +  a^, 
(a;  —  a)(a;  +  a),  etc.,  will  serve  as  a  solution  of  the  problem  of 

integrating   j  2  a;  dx. 

It  often  happens  that  different  methods  of  integration  lead 
to  different  results.  All  such  differences,  however,  can  occur 
only  in  the  constant  terms. 

For  example, 

f^Cx  +  ^ydx  =  8  f(x  + 1)^^(2^  + 1)=(3'  f  i)' 
=2^^.nx^-^iix-\-i. 


125.]  GENEBAL  PRINCIPLES   OF  INTEGRATION  201 

Integration  of  the  terms  separately  gives 

a  result  which  agrees  with  the  preceding  except  in  the  con- 
stant term. 

Again,  from  formula  XII, 


/— — -  =  tan~^a;,  or  —  cot~^a:. 


It  does  not  follow  from  this  that  tan~^a;  is  equal  to  —  cot~^a;. 
But  they  can  differ  at  most  by  an  additive  constant.  In 
fact,  it  is  known  from  trigonometry  that 

—  cot~^a?  =  tan~^a;  +  ^tt  -f  — , 

where  h  is  any  integer. 

In  a  similar  manner  the  different  results  in  formulas  XI 
and  XIII  can  be  explained. 

EXERCISES 

Integrate  the  following : 


.    \Vxdxi=  \x^ dx\.  8.   J  {ax  +  by  dx, 

J  J  x-\-  a 


3     r_^.  10.    fl^LzJEl^ 
J  ^ *  J    lax  —  X 

rmx-^-\  11     fsec^arc^a: 
*•    \ — -^dx.  •   J     tanx 

^    Vx 

J  12     r  siti  X  dx 
(a^  —  x^Y  dx.  '^  1  +  cos  X 

W',  6.    p^'-3^+ldx.  .        13.    f_^^('=fJI.) 

/t" —-'''/  J  x^  -^  X  log  x\     J  log  x/ 

7.   ^x^x'^  +  a^ydx.  14-   J^m* 


dx 


202  INTEGRAL  CALCULUS  [Ch.  L 

,15.  Jtanarrfyf^-J'^^^^^A         19.    ^ {a -^  h)^+^ dx, 

'^,16.  ^cotxdx.  20.   JcosHarrfxf^J^^t^^^c^xY 

17.  Xe^dx.  21.   J  sin  (m  +  n)a:c?a:. 

"^     18.  j  e==^  X  dx.  22.   J  sin  a;^  .  a;  dx. 

23.    j  cos^arc^xf  =  \  cosa:(l  —  sin^a;)^^]. 

24.  j  sin*a;<fa:.  25.    |sin2a:6fa;. 


V"' 


27 


^6.    Ttan^  a:  c?a:  r  =  ("(sec^  a;  -  1)  <^a;l . 

I  tan^  X  sec^  x  dx.  28.    |  cosec^  (aa;  +  6)  rfa. 

'^         29.    i  Vcot  a?  cosec^  a:  rfa:. 

30     r        <^3:        /_  C^ec'^xdxY 
J  sin  a;  cos  x\     ^    tan  a;    / 


^ 


.    J  sec*MtanM<iM. 

C     .      J   ( _  fcosec  u  cot  udu\ 
'J  \     ^         cosecM        / 


_       rtan  M  r?M 
J     sec  M 


f    '^  =  C 


35. 
36 
37 


f^^^-.  38.    f        ^^       (=f- 


1^ 

dx 


C     du  39.    f 

f      ^^  40.    f 

^  a*  +  />%2  -^  xV^« 


(ar-l)Va:2-2x 


125-126.]     GENERAL   PRINCIPLES   OF  INTEGRATION  203 

126.  Integration  by  parts.  If  u  and  v  are  functions  of  a?, 
the  rule  for  differentiating  a  product  gives  the  formula 

d  (uv)  =  V  du  +  u  dv, 

whence,  by  integrating  and  transposing  terms, 

\udv  =  uv  —  \vdu. 

This  formula  affords  a  most  valuable  method  of  integra- 
tion, known  as  integration  by  parts.  By  its  use  a  given 
integral  is  made  to  depend  on  another  integral,  which  in 
many  important  cases  is  of  simpler  form  and  more  readily 
integrable  than  the  original  one. 


Ex.1. 

Jlog 

xdx. 

Assume 

' 

u  - 

=  log  a:, 

dv  =  dx. 

Then 

du: 

dx 

V  =  X. 

By  substituting 

in  the  formula  for  in 

itegration  by  parts, 

Jloga 

:dx  = 

xlogx- 

-j-.. 

= 

a:  log  a:  ■ 

-X  =  x  (log  X  - 

-1) 

X 

=  X  (log  X  —  log  e)  =  a:  log  -. 


Ex.  2.  ixe^'dx. 

Assume  u  =  x,      dv  —  eFdx. 

Then  du  =  dx,      v  =  e*, 


and 


j  xe'^dx  =  xc*  —  i  e'^dx  =  e*(a;  —  1). 


Suppose  that  a  different  choice  had  been  made  for  u  and  dv  in  the 
present  problem,  say 

u  =  e*,     dv  =  X  dx. 


204  INTEGRAL   CALCULUS  [Ch.  I. 

From  this  would  follow 

du  =  e'dx,     y  =  — , 
2 

and  J  xe'dx  =  ^  arV  —  \  ^e*dx. 

—  e*dx  is  less  simple  in 

fomi  than  the  original  one,  and  hence  the  present  choice  of  u  and  do 
is  not  a  fortunate  one. 

No  general  rule  can  be  laid  down  for  the  selection  of  u  and  dv. 
Several  trials  may  be  necessary  before  a  suitable  one  can  be  found. 

It  is  to  be  remarked,  however,  that  as  far  as  possible  dv  should  be 
chosen  in  such  a  way  that  its  integral  may  be  as  simple  as  possible, 
while  u  should  be  so  chosen  that  in  differentiating  it  a  material  sim- 
plification is  brought  about.  Thus  in  Ex.  1,  by  taking  u  =  log  a:,  the 
transcendental  function  is  made  to  disappear  by  differentiation.  In 
Ex.  2,  the  presence  of  either  x  or  e*  prevents  direct  integration.  The 
first  factor  x  can  be  removed  by  differentiation,  and  thus  the  choice 
u  =  X  is  naturally  suggested. 


Ex.  3.  Kx^a'dx. 


From  the  preceding  remark  it  is  evident  that  the  only  choice  which 
will  simplify  the  integral  is 

u  =  x^,  dv  =  a'^dx, 

qX 

Hence  du  =  2x  dx,       v  = , 

log  a 

and  (x^a'dx  =  ^^  -  -^  (xa'dx. 

J  log  a     log  a  J 

Apply  the  same  method  to  the  new  integral,  assuming 

M  =  x,       dv  —  a'dXf 

whence  du  =  rfx,      v  = , 

log  a 

and  (xa'  dx  =  -^^  -  -^  f  o-rfar 

J  log  a     log  a  J 


logrt      {\o%ay 
By  substituting  in  the  preceding  formula, 

J  logaU         logo      (log a)'' J 


126-127. J      GENERAL   PRINCIPLES   OF  INTEGRATION  205 


EXERCISES 


1.    (Birr'^xdx.  7.    Kxcoi-'^xdx, 

J  ^    ^  8.   J  a:  SI 


2 

3.  \  x"^  cos  X  dx. 

4.  j  x^  log  X  dx. 

5.  (x^t&n-^xdx.  ^^'   J 


'^ 


6. 


sin  3  X  dx, 
\  e*  cos  X  dx, 
e"  sin  X  dx. 
j  sec  a;  tan  a:  log  cos  a:  rfa:.  11.    i  cos  a:  cos  2  a;  rfa:. 


127.  Integration  by  substitution.  It  is  often  necessary  to 
simplify  a  given  differential  f'(x)dx  by  the  introduction  of 
a  new  variable  before  integration  can  be  effected.  Except 
for  certain  special  classes  of  differentials  (see,  for  example, 
Arts.  138,  139)  no  general  rule  can  be  laid  down  for  the 
guidance  of  the  student  in  the  use  of  this  method,  but  some 
aid  may  be  derived  from  the  hints  contained  in  the  problems 
which  follow. 


Ex.1. 


J 


xdx 


Va2  -  a;2 

Introduce  a  new  variable  z  by  means  of  the  substitution  a'^  —  x^  =  z. 
Differentiate  and  divide  by  —  2,  whence  xdx  = Accordingly 

The  details  required  in  carrying  out  this  substitution  are  so  simple 
that  they  can  be  omitted  and  the  solution  of  the  problem  will  then  take 
the  following  form : 

r_^d^  =  ((a^-x^r^xdx  =  -  ^  ((a^-x^rH-2xdx)  =  -  (a^-x^^. 

In  this  series  of  steps  the  last  integral  is  obtained  by  multiplying  inside 
the  sign  of  integration  by  -  2  and  outside  by  -  J,  the  object  being  to 


206  INTEGRAL   CALCULUS  [Ch.  I. 

make  the  second  factor  the  differential  of  a^  —  x\  Thinking  of  the 
latter  as  a  new  variable,  the  integrand  contains  this  variable  affected  by 
an  exponent  {—  \)  and  multiplied  by  the  differential  of  the  variable,  in 
which  case  formula  I  can  be  applied. 

Ex.  2.     (^^^dx. 

J     X 
Assume  log  a;  =  2. 

Then  —  =  dz, 

and  |l2^rf.  =  pI  =  |  =  (!o|£l^ 

Here  again  it  is  not  necessary  to  write  out  the  details  of  the  substitu- 
tion, as  it  is  easy  to  think  of  log  a;  as  a  new  independent  variable  and  to 
perform  the  integration  with  respect  to  that.  It  is  then  readily  seen 
that  the  expression  to  be  integrated  consists  of  the  variable  logx  mul- 
tiplied by  its  differential  — ,   and   that  the  integration  is  accordingly 

X 

reduced  to  an  immediate  application  of  the  first  formula  of  integration. 
Thus 


Ex.3.        fgtan-^x      dx 

J  1  +  X2 


GoK^y 


logx  '  d(logx)z=       ^ 


Think  of  tan-^  a:  as  a  new  variable  and  apply  formula  IV.     Thus 


Ex.4.     r^iBli: 


+ 
'sin"^  X  dx 


dx 
Think  of  sin-^  a:  as  a  new  variable  and  — — ^^^  as  the  differential  of 

that  variable.     Apply  formula  I.  ^^  ~  ^* 

Ex.5.     j'(a;2+2a;  +  3)(x  +  l)«?x. 

Multiply  and  divide  by  2.     The  integral  then  takes  the  form 

i  JCar^  +  2  ar  +  3)  •  (2  x  +  2)dx. 

Observing  that  (2  a:  +  2)^a:  is  the  differential  of  ar^*  +  2  a:  -|-  3,  and  think- 
ing of  the  latter  expression  as  a  new  variable,  it  is  seen  that  formula  I  is 
directly  applicable,  leading  to  the  result 


127.]  GENERAL  PRINCIPLES   OF  INTEGRATION  207 

Ex.  6.     flog  cos  (x^  +  1)  sin  (x^  +1)'X  dx. 

Make  the  substitution 

ar2+l=2;. 

The  given  integral  takes  the  form 

J  j  log  cos  2;  siw  zdz. 

Make  a  second  change  of  variable, 

cos  z  =  y. 
Then  sin  zdz=:  —  dy. 

The  transformed  integral  is 

-lyf^gydy, 

to  which  the  result  of  Ex.  1,  Art.  126,  can  be  at  once  applied. 

It  will  be  observed  that  two  substitutions  which  naturally  suggest 
themselves  from  the  form  of  the  integrand  are  made  in  succession.  The 
two  together  are  obviously  equivalent  to  the  one  transformation, 

cos  (x^  +  1)  =  3/. 
Ex.7,     f    /^      . 

Either  put  x  =  az,  or  else  divide  numerator  and  denominator  by  a,  and 
write  in  the  form 


/ 


<^ 


v-(iy 


Regarding  -  as  a  new  variable,  this  comes  under  XI  and  gives  the 


result 


C       dx  •     ^  X  ,  ^ 

•^  Va2  -  x^  a 


=  -  cos-i-  +  C^. 


In  a  similar  manner  treat  Exs.  8-10. 

Ex.8,  r  ^^  . 

J  x'^  +  a^ 

Ex.9,     f— ^=. 

•^  xy/x^  -  a2 

Try  also  the  substitution  x  =  — 

z 


208  INTEGRAL   CALCULUS  fCn.  I. 

Ex.10,     f-        ^^ 


V2  ax  -  a;2 
Try  also  the  substitution  2:  -  a  =  x. 


Ex.  11.     •       ^^ 


/: 


Vx2  ±  a2 
Make  the  transformation 


From  this  follows,  by  differentiation, 


(1+     ^      -\dx  =  dz-r 


dx 


that  is,                        (v^2±a2  +  a;)       ""^       =  c?2, 
or,  •  = =  — • 

Ex.12,  r^^. 

Assume  x  —  a  _  ^ .    ^^^  is,  ar  =  a  — i-?» 

a:  +  a  1  —  « 

The  reasons  for  the  choice  of  substitution  made  in  this  and  the  pre- 
ceding example  will  be  made  clear  in  Arts.  133  and  139. 

Ex.13.     jcosecx£?x. 

Multiply  and  divide  by  cosec  x  —  cot  x.    It  will  be  readily  seen  that 

rdz 
the  integral  then  takes  the  form    \  — 

Another  method  would  be  to  use  the  trigonometric  formula 

sin  a:  =  2  sin  ^  cos  |, 

aec^ -dl-\ 
whence  I  cosec  xdx  =  I =  I  - 


• '  2  sin  ^  cos  r    *^        tan  ? 


:.  14.  J 


Ex.  14.    i  sec  a:  dx. 


Put  «  =  «  -  J,  and  use  Ex.  18. 


127-128.]     GENERAL  PRINCIPLES   OF  INTEGRATION  209 

Solve  the  problem  also  by  means  of  substitutions  similar  to  those 
used  in  the  preceding  example. 

ii^"^  Ex.  15.    f ^£ =  f i°^^ 

=  2f-5 P^ (s=2ax  +  6) 


2 


tan-i   ^«^  +  A-,  if  4  ac  -  52  >0. 
V4  ac  -  52  V4  ac  -  ¥• 

=  — ^::::r=r  log ' >   if  4  ac  —  6^  <  0. 

Vft-^  -  4  ac         2  aa;  +  6  +  V62  _  4  ac 

Ex.16,    f ^?£ l=f 2rf^ =f       ^(2^)        . 

^2a;2+2a;+3     ^  4^2+ 4a: +  6      ^(2a:+l)2+5 

In  this  form  it  is  ea^  to  integrate  by  taking  2  a:  +  1  as  a  new  variable. 

Ex.17,    f ^ . 

^3a:2-2a:+5 

Ex.18,    f i^^ . 

Ex.  19.  ^—^ ^ . 


V-9a;2+  30ar-24 
Ex.  20.    f  (3  a:  -  2)  cos  (3  a:  -  2)  dx. 


Ex.21,    f 


adx 


xVcfi  +  hx 


Substitute  y/a^  +  hx  —  z,  and  use  Ex.  12. 

\\)J  128.  Additional  standard  forms.  The  integrals  in  Exs. 
7-14  of  the  preceding  article,  and  in  Exs.  15-16  of  Art.  125, 
are  of  such  frequent  occurrence  that  it  is  desirable  to  collect 
the  results  of  integration  into  an  additional  list  of  standard 
forms. 


210  INTEGRAL   CALCULUS  [Ch.  I. 


du 

Va2  _  ^2  a      -  a 


_„     r      du  .,     1  u 

XT.    J— =:=r  =  6iii-i-,    or    -cos 


XVI.    f— z^^=log(i*  +  V^i2X^). 
XYII.    f_^  =  ltan-i^,    or    -icofi 

XIX.    f — ,^^      ,=^sec-*^,    or    -£:Cosec 


du  1         it*  1  1 «« 

,.=  =  — sec-*— ,    or    — cosec~*  — 

t*Vw2_«2     a  a  a  a 


XX.    i    ^  ==Yer8-^-- 


XXI.  j  tan  udu  =  -  log  cos  «*  =  log  sec  u, 

XXII.  j  cot  u  du  =  log  sin  u. 

XXI 11.  f  sec  udu  =  log  (sec  i*  +  tan  i*)  =  log  tan  (^  +  t)  • 

XXIY.  f  cosec  udu  =  log  (cosec  u  -  cot  u)  =  log  tan  ^  • 

129.  Integrals  of  the  form 

r  (Ax.  +  B)dx 
^  y/axi^  -\-bx  +  c 

Integrals  of  this  form  are  of  such  frequent  occurrence  as 
to  deserve  special  mention.  The  integration  is  readily  effected 
by  the  substitution  of  a  new  variable  which  reduces  the 
radical  to  a  simpler  form.  Two  cases  are  to  be  considered 
according  as  a  is  positive  or  negative. 

Case  I.  a  positive.  In  this  case  by  dividing  out  the 
coefficient  of  a^  the  radical  may  be  written 


^         a        a  ^\       'la  J  4a^ 


128-129.]      GENERAL   PRINCIPLES   OF  INTEGRATION  211 

The  given  integral  then  takes  the  form 

( Az  4-B—      \dz 
1    r*        (Ax-\-B)dx  -   ^    A  ^^J     (  _         ^ 


_  A    r  zdz 


V 


2  ,  -nac 
z^-\- 


P 


+ 


2aB-bA 


4:0,' 


C- 


dz 


^ac  —  W 


=  —Wax^  +  bx-[-  c-\ — log  X  +  - — \'\x^  -\--x-\-- 

^  2aVa        ""K       2a      ^         a       a 

Case  II.     a  negative.     When  a  is  negative,  by  dividing 
out  the  positive  number  —  a  the  radical  becomes 

^  a        a  4a2         \       2  a/ 

and  in  consequence  the  integral  takes  the  form 
1       y-         (Ax  +  B)dx 


^'^-hi-J 


V 


(Az+B-^)dz 
!_  A  2aJ 

—  a*^ 


—  -tao 


JblszA 

^     ■ia' 


V  2a 


_     A      r»  zdz 


^      -ia 


+ 


2  a^  -  6^ 


—  4:ac       2       2  aV 

<3 


-  0^^  n  dz 

\         A   „1 


V 


—  a^     4  a^ 


ac       2.2  a5  —  bA    '   _i 


4^2 

2a2 


ac       2 
z^ 


2  a  V  —  «  V^^  —  4  <xc 

-■^    /     2  ,   I     .       .   2aB-bA    .    _i    2a2;  +  6 
=  —  Vax^  +  bx-{-c-\ sin  ^  ■• 

^  2«V-a  V62-4ac 


212  INTEGRAL   CALCULUS  [Ch.  1. 

Ex.1.    f-(^^  +  ^)^^    . 
-^  V3  a;2  +  3  a;  +  2 

On  dividing  numerator  and  denominator  by  VS  the  integral  reduces  to 

—  X  H 1  dx 

a/8  a/8/ 


f 


which  by  means  of  the  substitution  a:  +  J  =  s  can  be  written 

•'         Vz2  +  ^  V3-'  -^  Vz2  ^  ^j 

=  J_  i5i±^il5  +  V3  log  (2  +  Vi2T5) 

=  _?_Va:2+a;  +  f +\/31og(a:  +  i  +  Va:2  +  x  +  |). 
V3 

Ex.2,    r       (2x  +  l)dx     _, 
•^V-2a;2-3a:-l 

Divide  numerator  and  denominator  by  v^.     The  integral  becomes 
^^±ldx 

=  -  V2  V^^Tia  _  _i_  sin-i  42 
2V2 

=  -  V-2x^-Sx^  1  -  -i-  sin-i  (4  z  +  3). 

2\/2 


130.   Imtegrals  of  the  form   j* 


da? 


(^05  +  B)  y/ax^  +  6aj  +  c 


Integrals  of  this  type  can  be  reduced  to  the  form  given  in 
the  preceding  article  by  means  of  the  reciprocal  substitution 


z 


129-130.]      GENERAL  PRINCIPLES  OF  INTEGRATION  213 

From  this  follow  the  relations 


and  Vaa^  '\-hx-\-  c  —  —  Vas^  ^  yg^  +  7, 

in  which  a  =  aW'  -  hAB  +  cA^, 

j3  =  -'2aB  +  hA, 

ry  —  a. 

When  these  expressions  are  substituted  in  the  given  inte- 
gral it  reduces  to 

-f      '^^      , 

which  has  the  form  discussed  in  Art.  129. 


EXERCISES  ON   CHAPTER   I 


^^+0^  +  1  ^^_ 


1.  f       ^^^       .  a  f-^       

•^  Va;2  +  2  a:  +  2  V  (^  +  1)  Va:^  +  2  a:  +  3 

2     r ^^ ,  [Divide  the  numerator  by  a;  +  l.J 

•^  xVo  a:2  —  4  a:  +  1 


^1  +  ^  +  ^?^^. 


3     r  C2a:-3)rfar,  9-   J ^ 

•^  V3  a:2  +  a;  -  2 

.     C    (4:x  +  5)dx  ^0-    1—7==^ 

4.    I     .^              ^ "^  a;Va;4  +  a;2  + 


V8  +  4  ar  -  4  a;2 

'■J 


1         -. 

a:  dx  [Assume  x  +  i-  =  s. ] 


V-ar2  +  2ar  +  l 

6.   (^jmdx. 

[Rationalize  the  numerator.] 


X 

11.    fe«*e*rfx 
'5  x^  dx 


12.     f5^1£E.        /      h 

J4  +  a;8      <t^f|AC-^ 


W^.  fV^rfi.  13.  fja+l^)^ 


T 


214 


"J 


dx 


INTEGRAL  CALCULUS 
25 


[Ch.  I.,1 


VI-  e^ 
[Put  e'  =  z.] 
dx 


15 


r       dx 

"I 


8x2 


■<Za: 


27 


(ix 


Vl  -a:* 


17.     f-^^. 


.If  

^•'xVl  -  log  a: 
C    e'^dx 

*       J   gX   _f.    g-X 


18.    1  X*  tan-i  a:  dx, 


Tx^  -  2  X  +  3 
J       a:2+l 

20.     f^^- 

21.  r  f  r-p-tM  ^''  -^ 

Jsin^+lL     J      cos2^        J 
22-   ll 


dO 


29     r COS  X  dx ^ 

•^  Vl  +  cos^  X  —  sin  X 


Cf      secx      y^i. 
J\a  — 6tanx/    -^JV* 


v5t 


{x—  a)  dx' 


Va*  -a^(x-  ay-  {x  -  a)* 


23.  r_*5 

J  a  + 


cost; 
tan  X  dx 


33.    I  vers^ ::• 

J  a  y/x 


24. 


i"rT 


6  tan^  x 
dx 


-]•; 


^x 


cot  X 


[Put  1  +  cotx  =  -.J 


a:2  V3  a:2  +  2  a:  +  1 
I  Substitute  a:  =  — J 
/        35.  Jlog  {X  +  \/^^^r^2)  ^.  '^ 


r 


-h 


CHAPTER  II 

REDUCTION  FORMULAS 

131.  In  Arts.  129, 130  the  integration  of  certain  simple  ex- 
pressions containing  an  irrationality  of  the  form  Vaa^  -{-bx-i-c 
has  been  explained.  As  was  shown  in  Art.  129,  the  radical 
can  be  reduced  to  the  form  -V  ±  a^  ±  a^  by  a  change  of  vari- 
able. It  remains  to  show  how  the  integration  can  be  per- 
formed in  such  cases  as,  for  example. 


CxW±  a^  ±  a^  dx,  C- 


x^dx 


V±  x^  ±  a2 

n  being  any  integer. 

For  this  purpose  it  is  convenient  to  consider  a  more  general 
type  of  integral  of  which  the  preceding  are  special  cases,  viz., 

^x'^ia  +  hx'^ydx,  (1; 

in  which  m,  w,  'p  are  any  numbers  whatever,  integral  or  frac- 
tional, positive  or  negative. 

It  is  to  be  remarked  in  the  first  place  that  n  can,  without 
loss  of  generality,  be  regarded  as  positive.  For,  if  n  were 
negative,  say  n  =—  n'^  the  integrand  could  be  written 

^""(^  +  4^)"  =  ^""f^"^"  ^  ^T  =  x'^-J'^'Xh  -h  ax»y. 

This  expression,  which  is  of  the  same  type  as  x^(a  +  hx^'y^  is 
such  that  the  exponent  of  x  inside  the  parenthesis  is  positive. 

215 


216  INTEGRAL   CALCULUS  [Ch.  II. 

It  will  now  be  proved  that  an  integral  of  the  type  (1)  can 
in  general  he  reduced  to  one  of  the  four  integrals 

(a)  A^x^'-'^i^a  -f  hx'^ydx,        (b)  Afx^'-^Xa  +  hx'^ydx, 

(c)  ^  J^r'^Ca  +  hsf'y-^dx,        (d)  A  Cx%a  +  hx'^y^Hx, 

plus  an  algebraic  term  of  the  form 

Bx\a  +  haf^y. 

Here  J.,  j5,  \,  /x  are  certain  constants  which  will  be  deter- 
mined presently. 

Observe  that  in  each  of  the  four  cases  the  integral  to 
which  (1)  is  reduced  is  of  the  same  type  as  (1),  but  that 
certain  changes  have  taken  place  in  the  exponents,  viz.,      "" 

the  exponent  m  of  the  monomial  factor  is  increased  or 
diminished  by  w, 

or,  the  exponent  p  of  the  binomial  is  increased  or  dimin- 
ished by  unity. 

The  values  of  \  and  /*  are  determined  by  the  following 
rule  : 

Compare  the  exponents  of  the  monomial  factors  in  the  given 
integral  and  in  the  integral  to  which  it  is  to  he  reduced.  Select 
the  less  of  the,  two  numbers  and  increase  it  by  unity.  The 
result  is  the  value  of  X.  In  like  manner^  compare  the  exponents 
of  the  binomial  factors  in  the  two  integrals^  select  the  less^  and 
increase  by  unity.      This  gives  fx. 

Thus,  if  it  is  desired  to  reduce  the  given  integral  to 
A  Cx"'-^(a  +  ba^ydx,     ' 
first  write  down  the  formula 
Cx^'Ca  -h  bx"ydx  =  A  Cx'^-^i^a  +  bx^ydx  -f  Bx^(a  -f-  b7f*y. 


131.]  BEDUCTION  FORMULAS  217 

The  exponents  of  the  monomial  factors  in  the  two  integrals 
are  m  and  m—  n  respectively,  of  which  m  —  yi  is  the  less. 
This,  increased  by  unity,  gives  the  value  of  \;  that  is, 
\  =  m  —  n  -{-1, 

Again,  the  exponent  of  the  binomial  factor  in  each  integral 
is  the  same,  namely  jo,  so  that  there  is  no  choice  as  to  which 
of  the  two  is  the  less.  Increase  this  number  p  by  unity  to 
obtain  the  value  of  /i.     Hence  {jl  =  p  -\-\, 

The  above  formula  may  now  be  written 

=  A^x'^-^ia  +  haf'ydx + Bx'''-^''\a  +  haf'y^^.   (2) 

In  order  to  determine  the  values  of  the  unknown  constants 
A  and  B^  simplify  the  equation  by  differentiating  both 
members.  After  dividing  by  x^~^(^a  +  hx^y  the  resulting 
equation  reduces   to 

x""  =  A-\-  Ba(m  —  n -\- 1)  +  Bh(m  +  np  -\-  l)a;". 

By  equating  coefficients  of  like  powers  of  x  in  both  members, 
the  values  of  A  and  B  are  found  to  be 

j^  —       ^(^  —  9^  +  1)  ^_ 1 

h(m  +  np  +  \y  h(m  +  np  +  1^ 

When  these  values  are  substituted  in  formula  (2),  it 
becomes 


fx'^^a  +  bsfydx 


=  ~  Tr~. ^^  )  ^      (^  +  hx'^ydx  +     w     .        ,  A    '    [a] 

Notice  that  the  existence  of  formula  (2)  has  been  proved 
by  showing  that  values  can  be  found  for  A  and  B  which 
make  the  two  members  of  this  equation  identical. 


218  INTEGBAL   CALCULUS  [Ch.  II. 

There  is  one  case,  however,  in  which  this  reduction  is 

impossible,  viz.,  when 

w  -f  wjt?  +  1  =  0, 

for  in  that  case  A  and  B  become  infinite.    [See  Ex.  4,  p.  221.] 
In  a  similar  manner  the  three  following  formulae  may  be 
derived : 


/' 


x"'(a  +  bx^ydx  ^ 

^     ^        fx  1  x"'+''(a  +  hx''ydx-\ \        ,/      .  fB] 

x%a  +  hsfydx 
w  +  wp  +  1^       ^  m  +  np  +  1 

=  —     ,       ^7^     I  x'^Ca  +  52;")^+^(^2; V-^ — ^^ [D] 

a/i(j9  +  l)     J      ^  ^  «n(jt?  +  l)  ■"   -' 


/■ 


The  cases  in  which  the  above  reductions  are  impossible 

are, 

For  formulae  [A]  and  [C],  when  m  +  np -^1  =  0; 

for  formula  [B]  ,  when  m  +  1  =  0 ; 

for  formula  [d]  ,  when  j9  +  1  =  0. 


Ex.1.     (xWa"^-  x^dx. 


K  the  monomial  factor  were  x  instead  of  x^,  the  integration  could 
easily  be  effected  by  using  formula  I.  Since  in  the  present  case  rn  =  3, 
n  =  2,  forjnula  [A],  which  diminishes  m  by  n,  will  reduce  the  above 
integral  to  one  that  can  be  directly  integrated. 

Instead  of  substituting  in  [A],  as  might  readily  be  done,  it  is  best  to 
apply  to  particular  problems  the  same  mode  of  procedure  that  was  used 
in  deriving  the  general  formula.  There  are  two  advantages  in  this. 
First,  it  makes  the  student  independent  of  the  formulas,  and  second, 
when  several  reductions  have  to  be  made  in  the  same  problem,  the  work 
ia  generally  shorter.     [See  Ex.  4.] 


131.]  BEDUCTION  FORMULAS  219 

Accordingly  assume 

(x\a'^  -  x^^dx  =  A(x(a^  -  x^)^dx  +  Bx%a^  -  x^)^, 

the  values  of  A  and  fj.  having  been  determined  by  the  previously  given 
rule. 

Differentiate,  and  divide  the  resulting  equation  by  x  (cfi  —  x^p.    This 

^^^®^  x^=A-{-B(2a^-6  x^), 

from  which,  by  equating  coefficients  of  like  powers  of  x, 

and  hence, 

(x^VaT^x^dx  =  —^(a^  -  x'^^xdx  -  \  x\a^  -  ar^)^ 

=  -  ^3(2  a2  +  3a;2)(a2  -  x'^)^, 
Ex.2.     fVa;2-2a;-3(/ar. 

By  following  the  suggestions  of  Art.  129,  this  integral  can  be  reduced 
to  the  form 

in  which  z  =  ar  —  1. 
Assume 

J(22  _  ^)\dz  =  A^(z^-  ^)-^dz  +  Bz(f  -  4)i 

In    determining   X   notice   that    m  =  0    in   both   integrals,    so   that 
X  =  0+1  =  1.     Also,  /A  =  -i+l  =  i. 

Ex.3.    (V2ax-x^dx. 

The  mode  of  procedure  of  Ex.  2  may  be  followed.     Another  method 
can  also  be  used,  as  follows : 

On  writing  in  the  form 

(xi(2a-x)^dxy 
and  observing  that 


vers 


-1 


X      C        dx 


f        ^^         =  f  ^-^(2a  -  x)-i  dx. 


«     ^  V2ax-x^ 


it  will  be  seen  that  the  integration  may  be  effected  in  the  present  case 
by  reducing  each  of  the  exponents  m  and  p  by  unity.  This  is  possible 
since  n  =  1  and  m  can  accordingly  be  diminished  by  1.     Hence  assume 

(x^(2a-  x)^  dx  =  A'(x-i(2a~  x)idx  +  B'xi(2 a  -  x)i. 


220  INTEGRAL   CALCULUS  [Ch.  II. 

The  exponent  of  the  binomial  in  the  new  integral  may  be  reduced  in 
turn  by  assuming 

(x-^(2  a-x)^dx  =  A "(x-^(2  a  -  xy^dx  +  B"x^i2 a  -  x)^. 

When  this  expression  is  substituted  for  the  integral  in  the  second 
member  of  the  preceding  equation,  the  result  takes  the  form 

f  y/2ax-x^dx  =  A  f        ^^         +  Bx^ {2  a  -  x)^  +  Cx^{2  a-x)\ 
J  -^  y/2ax  —  x^ 

in  which  A,  B,  C  are  written  for  brevity  in  the  place  of  A' A",  A'B",  B' 
respectively.  The  values  oi  A,  B,  C  are  calculated  in  the  usual  manner 
by  differentiating,  simplifying,  and  equating  coefficients  of  like  powers 
of  X. 

The  method  just  given  requires  two  reductions,  and  heilce  is  less 
suitable  than  that  employed  in  Ex.  2,  which  requires  but  one  reduction. 

The  rule  for  determining  the  values  of  X  and  fi  may  now 
be  advantageously  abbreviated.  Let  m,  p  be  the  exponents 
of  the  two  factors  in  the  given  integral,  and  m',  p'  the  corre- 
sponding exponents  in  the  new  integral.  Of  these  two 
pairs,  m,  p  and  m'^  p\  one  of  the  numbers  in  the  one  pair  is 
less  than  the  corresponding  number  in  the  other  pair.  This 
fact  will  be  expressed  briefly  by  saying  that  the  one  pair  is 
less  than  the  other  pair.  With  this  understanding  the 
preceding  rule  may  be  expressed  as  follows  : 

Select  the  less  of  the  two  pairs  of  exponents  m,  p  and 
m',  p'.  Increase  each  number  in  the  pair  selected  by  unity. 
This  gives  the  pair  of  exponents  X,  fi, 

Ex.4.    ^     ^'^^ 


^■^7^. 


(x2  +  a2)^ 
Assume  successively 

f  a:<(x2  +  a«)~*  dx  =  A'    (x*(x*  +  a^)"*  dx  +  B'x*(x^  +  a«)~*, 

^x*(x^  +  a^)'^dx  =  Af  jx^x^  +  a^yhx  +  B"x^(x^  +  a^)*, 

(x^(x^  +  a2)-i  dx  =  A '"  f  (z*  +  a2)~i  dx  +  B'"x(x^  +  a^)K 


131.]  REDUCTION  FORMULAS  221 

These  equations  may  be  combined  into  the  single  formula 
(x\x^  +  d^)~^  dxz=A({x'^+  a2)-i  dx  +  Bx  (x2  +  a2)i 

+  CxXx"^  +  a2)i  +  Dx^x"^  +  a2)~*. 
The  values  of  the  coefficients  are  found  to  be 


Hence 


•^  2Va;2-j-  a2 


In  this  example  three  reductions  were  necessary  ;  first,  a  reduction  of 
type  [1>J,  second,  and  third,  a  reduction  of  type  \A'\ .  Can  these  reduc- 
tions be  taken  in  any  order  ? 

The  different  possible  arrangements  of  the  order  in  which  these  three 
reductions  might  succeed  each  other  are 

(1)    [-4],  [^],  [D]  ;         (2)    [^],  [1>],  [^];         (3)    [1>],  [^],  [^], 

of  which  number  (3)  was  chosen  in  the  solution  of  the  problem.  Of 
the  other  two  arrangements,  (2)  can  be  used,  but  (1)  cannot.  For, 
after  first  applying  [^]  (which  would  be  done  in  either  case),  the  new 
integral  is 

( x\a^  +  x'^y^  dx. 

If  [^]  were  now  applied,  it  would  be  necessary  to  assume 

Ja:2(a2  +  x"^)"^  dx  =  ^  f  (a2  +  x'^)~^-\-  Bx{cfi  +  x'^)'^. 

This  equation,  when  difEerentiated  and  simplified,  becomes 

a:2  =  ^  +  5a2, 

a  relation  which  it  is  clearly  impossible  to  reduce  to  an  identity  by 
equating  coefficients  of  like  powers  of  x,  since  there  is  no  x^  term  in 
the  right  member  to  correspond  with  the  one  in  the  left  member.  It 
will  be  observed  that  this  is  the  exceptional  case  mentioned  on  page  218, 
in  which  m  +  np  +  1  =  0. 

Ex.  5.   Show  thaib  the  integral    fC^  -^  )  ^x  can  be  integrated  by 

four  reductions.  Prove  that  these  can  be  arranged  in  six  different 
orders,  and  determine  those  which  can  be  used. 


222                                  INTEGRAL   CALCULUS  [Ch.  II.  131. 

Ex.    6.    f — ^-— .  (T-Ex.  13.    (V^^Tadx. 

J  (a:'-'  +  iy  J 

Ex.    8.    (•_^!*L_.  ■^L^        .^..^ J      ^   V^ 

y^'       rt^x.    9.     iV^^^^dx,  Ov 

J^^                 ,        ..  Ex.16.    f_^=.    D>V 
•^  xWa'  -  x^ 

Ex.11,    f-i^^ /  Ex.17.    r,^f  ^o^«'    K^ 

Ex.  12.    f  (a2+  x2)^rfar.  Ex.  18.    f  Vl  -  2  x  -  x^ rfar.  ^ 


^' 


Ex.  19.   Show  that 


r       ^^       = 1 r ^ +(2n-3)f ^-^ 1 


''^^' 


l> 


CHAPTER   III 

INTEGRATION  OF  RATIONAL  FRACTIONS 

132.  Decomposition  of  rational  fractions.  The  object  of  the 
present  chapter  is  to  show  how  to  integrate  fractions  of  the 
form  ,  ^ 

wherein  (f)(^x)  and  '>jr(^x)  are  polynomials  in  x. 

The  desired  result  is  accomplished  by  the  method  of  sepa- 
rating the  given  fraction  into  a  sum  of  terms  of  a  simpler 
kind,  and  integrating  term  by  term. 

If  the  degree  of  the  numerator  is  equal  to  or  greater  than 
the  degree  of  the  denominator,  the  indicated  division  can  be 
carried  out  until  a  remainder  is  obtained  which  is  of  lower 
degree  than  the  denominator.  Hence  the  fraction  can  be 
reduced  to  the  form 

^  =  a."  +  5.«-  +  ...  +  ^, 

in  which  the  degree  oif(x')  is  less  than  that  of  '^(x)' 

As  to  the  integration  of  the  remainder  fraction  ^^  ^  ,  it  is 

to  be  remarked  in  the  first  place  that  the  methods  of  the 
preceding  articles  are  sufficient  to  effect  the  integration  of 
such  simple  fractions  as 

x—a   (x  —  ay^''      '    x^±d?''  (x^±a^y^^      '    x^-\-'mx-{-n 

Now  the  sum  of  several  such  fractions  is  a  fraction  of  the 
kind  under  consideration,  viz.,  one  whose  numerator  is  of 

223 


224  INTEGRAL   CALCULUS  [Ch.  III. 

lower  degree  than  its  denominator.  The  question  naturally 
arises  as  to  whether  the  converse  is  possible,  that  is :  can  every 
fraction  ^^^£2.  he  separated  into  a  sum  of  fractions  of  as  simple 

types  as  those  given  in  (1)  f 

The  answer  is,  yes. 

Since  the  sum  of  several  fractions  has  for  its  denomina 
tor  the  least  common  multiple  of  the  several  denominators, 

it  follows  that  if   •^^  ^    can  be  separated   into  a  sum  of 

simpler  fractions,  the  denominators  of  these  fractions  must 
be  divisors  of  'yjr(x^.  Now  it  is  known  from  Algebra  that 
every  polynomial  '^^(x)  having  real  coefficients  (and  only  those 
having  real  coefficients  are  to  be  considered  in  what  follows) 
can  he  separated  into  factors  of  either  the  first  or  the  second 
degree^  the  coefficients  of  each  factor  heing  real. 

This  fact  naturally  leads  to  the  discussion  of  four  different 
cases. 

I.  When  '>^(x')  can  be  separated  into  real  factors  of  the 
first  degree,  no  two  alike. 

E.g. ,  '^^(2?)  =  (x  —  a)(x  —  6)  (x  —  c). 

II.  When  the  real  factors  are  all  of  the  first  degree,  some 
of  which  are  repeated. 

III.  When  some  of   the   factors   are   necessarily  of  the 
second  degree,  but  no  two  such  are  alike. 

U.g.,    ylr^x)  =^Cx^  +  a^Ca^-hx  +  l)(rr -hXx-  c^. 

IV.  When  second  degree  factors  occur,  some  of  which 
are  repeated. 

Kg.,  f(x)  =  (:t-2  +  a2)V-^  +  0(^-^)- 


132-133.]    INTEGRATION  OF  RATIONAL  FRACTIONS  225 

133.   Case  I.     Factors  of  the  first  degree,  none  repeated. 
When  ylr(x)  is  of  the  form 

^fr(x')  =  (x  —  d)(x  —h}(x—  6)  •••  (x  —  7l), 

assume 


i|r(a;)      X  —  a      x  —  h      x  —  c  x  —  n 

in  which  A^  B^  C^  '"^  N  are  constants  whose  values  are  to  be 
determined  on  condition  that  the  sum  of  the  terms  in  the 
right-hand  member  shall  be  identical  with  the  left-hand 
member. 

Ex.l.     (tpl^d^. 

Dividing  numerator  by  denominator,  -i— -  =  x , 

^  ^  x2-3a:  +  2  x^~^x+2 

Assume  -^ ^  =  ^d_+    ^ 


(a:-l)(a;-2)      x-1     x-2 

By  clearing  of  fractions, 

(1)  x  =  A(x-2)  +  B(x-iy 

In  order  for  the  two  members  of  this  equation  to  be  identical  it  is 
necessary  that  the  coefficients  of  like  powers  of  x  be  the  same  in  each. 

Hence  1  =  A-^B,  0  =  -2A-B, 

from  which  ^  =  —  1,  B=2. 

Accordingly  the  given  integral  becomes 

Kx  +  ^^-^)dx=:aL+\og(x-l)~2log(x--2)'rC 

A  shorter  method  of  calculating  the  coefficients  can  be  used.  Since 
equation  (1)  is  an  identity,  it  is  true  for  all  values  of  x.  By  giving  x 
the  value  x=l  the  equation  reduces  to  1  =  ^(— 1),  or  A=:  —  l. 
Again,  assume  x  =  2.    Whence  2  =  B. 


226     JtcVf  ^     -^INTEGRAL   CALCULUS       ^^K    '"^    [Ch.  IH 

Ex.  2M  ^^  Ex.10    ri?^±il^.     V 

^a;2-a2  J  2x2  +  3  3:- 2 

Ex.3,    flu^dor.  Ex.11.    f-M+^&l^^.  lAL 

J  x^-x  X  ^  a;  (a-- -a)  (a: +  6)    ^ 


Ex.4 


Cix^-\2)dx  ^    Ex.  12.  r_i£±ii^.      \i 

Ja;2+4a:+3  J2x-a:2-a;8       ,  l^ 


Ex.6. 


f        a:</a;  Ex.14,    f ^ ^^ 

)x^-4.x  +  i  Ja^x^-b^ 

Ex  7     r_i£!^iM£_  Ex.15,    f       (^^-^-1)^. 

J(a:2-4)(4i2-l)"  J(x2-3x+2)V23r« 

P^  «     ra:2-2ca:  +  qc-a&+6c^^         [Separate     f~^~\   into  par- 
^^-  «:  J  ^,-aXx-b)(x-c)'^''  tialfractions.]^'-  3  ^+  2 


(a;2-4)(4a;2-l) 
Ex.8,    f 


Ex.9. 


fx2(.+a)-i(:.+6)-i^..      Ex.16,    f (^^+^^-^^ 


Vx2+4a:+7 

134.    Case  II.     Factors  of  the  first  degree,  some  repeated. 

Ex  1     rC5a:2-3ar+  l)</x. 
■  J  a:  (x  -  1)8 

Assume 
rn  5a:2-3a:4  1^^         BCD 

^^  a:(x-l)8  X      a:- 1      (a; -1)2"^  (x- 1)8* 

To  justify  this  assumption,  observe  that : 

(a)  In  adding  the  fractions  in  the  right-hand  member,  the  least  com- 
mon multiple  of  the  denominators  will  be  x  (x  —  ly,  which  is  identical 
with  the  denominator  in  the  left-hand  member. 

(6)  Further,  the  expressions  a:,  a:  —  1,  (a:  —  1)^,  (x  —  1)8  are  the  only- 
ones  which  can  be  assumed  as  denominators  of  the  partial  fractions, 
since  these  are  the  only  divisors  of  x(x  —  1)8. 

(c)  When  equation  (1)  is  cleared  of  fractions,  and  the  coefficients  of 
like  powers  of  x  in  both  members  are  equated,  four  equations  are  ol)- 
tained,  which  is  exactly  the  right  number  from  which  to  determine  the 
four  unknown  constants  A,  B,  C,  D. 

Instead  of  the  method  just  indicated  in  (c)  for  calculating  the  coeffi- 
cients, a  more  rapid  process  would  be  as  follows : 

By  clearing  of  fractions,  the  identity  (1)  may  be  written 

6  a;«  -  3  X  +  1  =  ^  (x  -  1)8  -h  Bx  (x  -  1)2  +  Cx  (x  -  1)  +  7)a;. 


133-134.]     INTEGRATION  OF  RATIONAL  FRACTIONS  227 

Putting  a;  =  1  gives  at  once 

3  =  i>. 

Substitute  for  D  the  value  just  found,  and  transpose  the  correspondmg 

term.     This  gives 

b  x'^  -  Q  X  -t  I  =  A  {x  -  ly  -{-  Bx  {x  -  \y  +  Cx  {x  -  1). 

It  can  be  seen  by  inspection  that  the  right-hand  member  of  the  result 
is  divisible  by  a?  —  1.  As  this  relation  is  an  identity,  it  follows  that  the 
left-hand  member  is  also  divisible  by  x—\.  When  this  factor  is  re- 
moved from  both  members,  the  equation  reduces  to 

bx-  \  =  A(x-\y-\-Bx(x-l)+  Cx. 

Now  put  a;  =  1.     Then 

C  =  4. 

Substitute  the  value  found  for  C,  transpose,  and  divide  by  a:  —  1. 
The  result  is  \  =  A{x-l)  +  Bx. 

By  giving  x  the  values  0  and  1  in  succession,  it  is  found  that 

A  =  -l,        5  =  1. 
Accordingly, 

r(5x^-3x+l)dx^  rf_l  +  _j_  +  __^_  +  _3_U 

J       x{x-iy        J\    X    x-i    (x-iy    (x-iy) 

=  log.^-l        ^^-^ 


X         2(x  -  1)2 

Ex.    2.     (—J^ J     Ex.    9.     r^£!±^^!±r«±l}^±«rf^. 

J(x^iyix+1)  J              x\a+x) 

Ex.    3.     rC^^- 11^+26)^0:.  ^^^^^     nx^-l)dx, 

^'^'    *•    ^{x^-a^y  Ex.ll.'|(ax2+6;r8)-irfx.'\ 

Ex.    5.     r(^2a:+l)rfa;,  ^^  ^^      f      ix^-x'^-r)dx       . 

•^    x\x  -f  V2)a  '           J  (a;  _  1)2  Va:^  -  2  a:  +  2 

^^-    ^-    f-^i^fl^^-  [Separate   ^Izz^iz-l   into  partial 

/Ex  7   r^{^i±_«Mi!^ 

J  a:4  -  2  a%2  +  a* 
yEx.    B.    I  ^2rf. 


N^ 


fractions.] 

Ex.13.     {       ^'       rfa?. 
-^  (a;  —  a)8 

(2  -I-  v^  -  V^  a:)8  [Substitute  a:  -  a  =  «.] 


'm- 


228  INTEGRAL   CALCULUS  [Ch.  III. 

135.  Case  III.  Occurrence  of  quadratic  factors,  none 
repeated. 

'  •     •   J(x2+l)(a;2+2a;+2)* 
Assume 

(Vi  4a;2  +  5a:  +  4         _  Ax  ■{■  B         Cx-{-  D 

^^  (a;2+l)(a:2  +  2x+2)       x^^-1       x^-\-2x^-2 

Then 

(2)  4a;«  +  5a:4-4  =  (^a:  +  ^)(a;2  +  2a:  +  2)  +  (Ca:  +  7))(a:2+l). 

By  equating  coefficients  of  like  powers  of  x 

0  =  J+C,  5  =  2^  +  2£  +  C, 

4  =  2^+5+ A  4  =  25  + A 

from  which 

^  =  1,  5  =  2,  C  =  -l,  i)  =  0. 

Hence  the  given  integral  becomes 

C{x  +  2)dx_C      xdx       ^2tan-ia;+tan-i(a:+l)+ilog  /'+^     . 

To  make  clear  the  reasons  for  the  assumption  which  was  made  con- 
cerning the  form  of  equation  (1),  observe  that  since  the  factors  of  the 
denominator  in  the  left  member  are  a-^  +  1  and  a;^  +  2  a:  +  2,  these  must 
necessarily  be  the  denominators  in  the  right. member.  Also,  since  the 
numerator  of  the  given  fraction  is  of  lower  degree  than  its  denominator, 
the  numerator  of  each  partial  fraction  must  be  of  lower  degree  than  its 
denominator.  As  the  latter  is  of  the  second  degree  in  each  case,  the 
most  general  form  for  a  numerator  fulfilling  this  requirement  (i.e.,  to  be 
of  lower  degree  than  its  denondnator)  is  an  expression  of  the  first  degree 
such  as  Ax  +  J5,  or  Cx  +  B. 

Notice,  besides,  that  in  equating  the  coefficients  of  like  powers  of  x  in 
opposite  members  of  equation  (2),  four  equations  are  obtained  which 
exactly  suffice  to  determine  the  four  unknown  coefficients  A,  B,Cf  D. 

1     (Adx_,  E^.6     C{^x-^)dx^ 

'  J  x^  +  ^x  J     x^  +  2x^ 


Ex.2. 


Ex  3     f  ^^^  Ex  7     f      ^^^ 

"  ^(a;+l)(a;2+l)  '    '  J  x^ -^  x'^  ■\- 1 

Ex.5,    f      {a'-^')dx      .  E,.9.    r     {^  +  ^^x  +  2)dx    . 


i\\x}^ 


135-136.]     INTEGRATION   OF  RATIONAL   FRACTIONS  229 

136.  Case  IV.  Occurrence  of  quadratic  factors,  some 
repeated.  This  case  bears  the  same  relation  to  Case  III 
that  Case  II  bears  to  Case  I,  and  an  exactly  analogous  mode 
of  procedure  is  to  be  followed. 


J  (a;2  +  2)8 


Ex 

Assume 

,j.  2x^-  x^  i-  8x^  +  4:  ^  Ax -{-  B       Cx  +  D        Ex  +  F 

^  ^  (:r2  +  2)3  a:2  +  2        (a;2  +  2)^  ^  {x^  +  2)3* 

Whence,  by  clearing  of  fractions, 

2 x^  -  x^  +  8  x^  +  4:  ={Ax  +  B){x'^  +  2)2  +  (Ca:+  D){x^  +  2)+Ex-\-  F. 

Instead  of  equating  coefficients  of  like  powers  of  x,  as  might  be  done, 
the  following  method  of  calculating  the  values  of  J.,  ^,  C,  •••  is  briefer. 

Substitute  for  x^  the  value  —  2,  or,  what  is  the  same  thing,  let 
x  =  V—  2.  This  causes  all  the  terms  of  the  right  member  to  drop  out 
except  the  last  two,  and  equation  (1)  reduces  to 

_  S\/^^  =  EV^  +  F. 

By  equating  real  and  imaginary  terms  in  both  members, 

-  8  =  ^,    ^  =  F. 

Substitute  the  values  found  for  E  and  F  in  (1),  and  transpose  the 
corresponding  terms.  Both  members  will  then  contain  the  factor  x^-^2. 
On  striking  this  out  the  equation  reduces  to 

2a:8-a:2+4a:  +  2  =  (^Ax  +  B^{xP-  +  2)+  Ca:  +  2). 

Proceed  as  before  by  putting  x^  =  —  2.     Whence 

4=CV^r2  +  D, 

and  therefore  0  =  C,    4  =  Z). 

Substitute  these  values,  transpose,  and  divide  by  a:^  +  2.     This  gives 

2x-\  =  Ax-\-  B, 

whence  ^  =  2,    5  =  —  1. 

The  given  integral  accordingly  reduces  to 

"^xdx 


J  a:?  4-  2  J  (x^  +  2)2     J  {x 


(a;2  +  2)2     J  {x^  +  2)8 


230  INTEGRAL   CALCULUS  [Ch.  III.  136-137. 

The  first  term  becomes 

-  The  second,  integrated  by  the  method  of  reduction  (Chap.  II),  gives 
^      +_l.taii-i^ 


x^  +  2  V2           y/2 

Finally,  by  applying  formula  I  the  last  term  integrates  immediately 
Hence 

J           {x^  +2)8  -  ^  '^     ^  ^  ^  a:2  +  2  ^  (x2  +  2)2 

Ex.  2.    a^Y  dx.  Ex.  5.    f(-^/  +  V^. 

J\x2  +  1/  J     a:2(x2+l>2 

>v   E^.  3.    C{x  +  ay+a^  ^^^  ^x.  6.    Cl^±l^^^^sU^  dx. 

V     Ex.4,    f ^^^ Ex.7,    f     ^'^^    . 

\                 J(l  +  a:)(l  +  ar2)2  J  (1  +  a:2)8 


The  principles  used  in  the  preceding  cases  in  the  assump- 
tion of  the  partial  fractions  may  be  summed  up  as  follows ; 

^ach  of  the  denominators  of  the  partial  fractions  contains 
one  and  only  one  prime  factor  of  the  given  denominator.  When 
a- repeated  prime  factor  occurs.,  all  of  its  different  powers  must 
he  used  as  denominators  of  the  partial  fractions. 

The  numerator  of  each  of  the  assumed  fractions  is  of  degree 
one  lower  than  the  degree  of  the  prime  factor  occurring  in  the 
corresponding  denominator. 

137.  General  theorem.  —  Since  every  rational  fraction  can 
be  integrated  by  first  separating,  if  necessary,  into  simpler 
fractions  in  accordance  with  some  one  of  the  cases  considered 
above,  the  important  conclusion  is  at  once  deducible  : 

The  integral  of  every  rational  fraction  can  he  found.,  and  is 
expressihle  in  terms  of  algehraic,  logarithmic,  and  inverse-trigo- 
nometric  functions. 


CHAPTER   IV 
INTEGRATION  BY  RATIONALIZATION 

At  the  end  of  the  preceding  chapter  it  was  remarked  that 
every  rational  algebraic  function  can  be  integrated.  The 
question  as  to  the  possibility  of  integrating  irrational  func- 
tions has  next  to  be  considered.  This  has  already  been 
/touched  upon  in  Chapter  II,  where  a  certain  type  of  irra- 
tional functions  was  treated  by  the  method  of  reduction. 

In  the  present  chapter  it  is  proposed  to  consider  the 
simplest  cases  of  irrational  functions,  viz.,  those  containing 
^ax  +  h  and  -\/ ax^  -\-hx  -\-  c^  and  to  show  how,  by  a  process 
of  rationalization,  every  such  function  can  be  integrated. 

138.  Integration  of  functions  containing  the  irrationality 
\/aa?  +  6.  When  the  integrand  contains  ^ ax  -f-  5,  that  is, 
the  wth  root  of  an  expression  of  the  first  degree  in  x^  but  no 
other  irrationality,  it  can  be  reduced  to  a  rational  form  by 
means  of  the  substitution 


Ex.1. 

le 

dx 

p  +  3-1 

ASSUDC 

that  is, 
Then 

and 

V2a;  +  3  =  2, 
2a;+3  =  z2. 

dx  =  z  dz, 
C          dx            _Czdz          -^^^^^^     1^ 

•^V2a:  +  3-l      *^^-l 

=  V2  a;  +  3  +  log  (  V2  a;  -f  8  -  1). 
231 

232  INTEGRAL   CALCULUS  [Ch.  IV. 

Ex.2.  Jl-f^"-^"- v^^^^ 


^.      t:»_  «     r  1  +  a;6_—  a;s  —  v^ 
a:*  +  a; 


It  would  appear  at  first  sight  that  this  integrand  contains  several 
irrationalities,  viz.,  Vx,  Vx,  Vx,  It  is  readily  seen,  however,  that  they 
are  all  powers  of  Vx,  and  hence  the  substitution  Vx  =  z  will  rationalize 
the  expression  to  be  integrated.  iryJ 

tK     Ex.3,    f       ^^      .^y^^^^l       ()^Ex.6.    f— ^^ tvL^^^''        ^ 


Ex  4    f     ^^     -  Ex.7,  r ^^    ^       ♦>  A'^' 

Ex.  5.    f ^? Ex.  8.    f  4^^^- 


/tH 


When  two  irrationalities  of  the  form  Wax  +  5,  Vc^+^ 
occur  in  the  integrand,  the  first  radical  can  be  made  to  dis- 
appear by  the  substitution 


}  "Vax  +  5  =  2. 

The  second  radical  then  reduces  to 


^  a 
r\       and  the  method  of  the  next  article  can  be  applied. 


y^-^ 


Integration  of  expressions  containing  Vax^  -\-bx  +  c. 
Every  expression  containing  ^ax^  +  bx  +  c^  but  no  other 
irrationality,  can  be  rationalized  by  a  proper  substitution. 
In  order  to  make  the  necessary  steps  clearer,  a  geometrical 
interpretation  of  the  problem  will  be  very  useful. 

To  this  end  let  the  given  radical  be  represented  by  y; 
that  is,  let 

i/^  =  aa^-^bx  +  o,  (1) 


138-139.]        INTEGRATION  BY  RATIONALIZATION 


233 


If  now  (a;,  ^)  be  regarded  as  the  rectangular  coordinates  of 
a  point  in  a  plane,  equation  (1)  represents  a  conic  (Fig.  60). 

Let  (A,  k'),  or  Q,  be  a  given 
point  on  this  curve.  The  equa- 
tion of  any  line  through  this 
point  is 

i/-k  =  z(x-h),        (2) 


X 


Fig.  60. 


in  which  z  is  the  slope  of  the 

line.     The  line  (2)  will  inter-     

sect  the  conic  in  a  second  point 

P.     It  is  geometrically  evident 

that  the  coordinates  (rr,  «/)  of  P  depend  on  the  value  of  2, 

and  in  such  a  way  that  to  each  value  of  z  corresponds  only 

one  pair  of  values  x^  y. 

Consequently  the  variables  x  and  y  can  be  rationally 
expressed  in  terms  of  the  variable  z.  This  is  done  by  treat- 
ing equations  (1)  and  (2)  as  simultaneous,  and  solving  for 
X  and  y  in  terms  of  z. 

For  example,  suppose  it  were  desired  to  rationalize  an 
expression  containing  Vic^  _  5  ^j  ^  g. 


Let 


^2=a;2_5a.^.8, 


and  select  (1,  2)  for  the  point  Q, 

Then  y-1  =  z{x-V) 

represents  any  line  passing  through  Q.  In  solving  these 
two  equations  simultaneously  for  x  and  y^  the  elimination 
of  y  gives 

z\x-Vf^-^z{x-r)  =  ^-bx^\, 

This  quadratic  equation  in  x  has  two  roots,  one  of  which 
should  be  a;  =  1,  since  this  is  the  value  of  x  at  C  one  of  the 


234  INTEGRAL   CALCULUS  [Ch.  IV. 

points  of  intersection.     The  other  root,  corresponding  to  the 
variable  point  P,  is 

22—42  —  4 

X  = • 

22-1 

From  this  follows 

or  y  =  V;»^-5^  +  8=-'^^'-^^-^- 

Two  particular  cases  of  the  method  given  above  deserve 
to  be  noticed. 

(a)  When  the  conic  intersects  the  x-axis. 

In  this  case  the  quadratic  expression  aoc^  +  hx-\-c  has  real 

factors,  say, 

ax^  -\-hx-\-  e  =  a(x  —  a) (a:  —  /3). 

The  conic  (1)  intersects  the  a;-axis  in  the  two  points 
(a,  0)  and  (y8,  0),  either  one  of  which  may  be  conveniently 
selected  for  the  point  Q. 

The  equation  of  any  line  QP  through  the  first  point  is 

y  =  z(x-a^,  (^) 

and  the  equation  of  any  line  through  the  second  point, 

y  =  z(x-^y  (A') 

Either  one  of  these  equations,  combined  with  (1),  will 
effect  the  desired  rationalization. 

(h)   When  the  conic  is  an  hyperbola. 

This  case  occurs  when  the  coefficient  of  a^  is  positive. 
The  curve  extends  to  infinity  in  two  different  directions, 
namely,  the  directions  of  the  asymptotes.  If  one  of  the 
points  at  infinity  on  the  curve  be  taken  for  the  point  Q,  the 
lines  QP  passing   through  this  point  are  parallel   to  that 


139. J  INTEGRATION  BY  RATIONALIZATION  235 

asymptote  which  touches  the  curve  at  Q.     The  equations  of 
the  asymptotes  are 

Accordingly  the  lines  parallel  to  the  one  asymptote  are 

and  those  parallel  to  the  other 

Either  of  these  equations  used  in  place  of  (2)  will  serve 
equally  well  in  expressing  x  and  y{=-^a^ -\-hx-\-  c)  ration- 
ally in  terms  of  a  new  variable  z, 

Ex.l.   ^  ^^ 


(x  +  V ar2  +  2  ar  -  1)^ 


The  conic  y  =  y/x^  +  2  a:  —  1  is  an  hyperbola  and  formula  (B)  can  be 
applied.     This  gives 

Vx^  +  2  a:  -  1  =  x  +  Zy 

whence  by  squaring  and  solving  for  x, 

z^+1 
^-2(l-z)' 
and  accordingly 

Vx  +2x      1-       2^^_^^ 
When  these  expressions  are  substituted  in  the  given  integral,  it  becomes 

=  i[-^  +  41og(l  +  .)  +  ^] 

=  l(x-y/x^+2x-l)  + +21og[H-Vrr2+2a;-l-a:]. 

l_a:+Va;2+2ar-l 


Since  the  conic  y  =  Vx^  +  2  a:  —  1  cuts  the  ar-axis,  formula  (A)   [or 
(^')]  could  be  used  for  the  purpose  of  rationalization. 


236  INTEGRAL   CALCULUS  [Ch.  IV. 


Ex 


-•/: 


\/\  +  xdx 


The  denominator  being  rationalized,  the  integrand  takes  the  form 


vT^^^ 


(1-xy 
The  conic 

y  =  VI  -  a:2 

intersects  the  a:-axis  in  two  points  (i  1,  0). 

If  the  point  (1,  0)  be  chosen  for  Q,  the  equation  of  any  line  passing 
through  this  point  is 

y  =  z{x-  1). 

The  simultaneous  solution  of  these  two  equations  gives 

22  _  1  _2z 

whence  fVl^^,  ^  f- 2.^. 

=  2(—  2  +  tan-^2) 

\     a:  —  1  X  —  \    I 


Ex.  3. 


J 


<fx 


(l_a:)(l-Vl-a:2) 


Ex.4,  r  ^^ 


V2  a:2  _  .3  a;  +  1  [V2  x2  -  8  a:  +  1  +  v^(a:  -  1)] 

140.  From  what  precedes,  combined  with  the  theorem  of 
Art.  137,  it  follows  that  every  rational  function  depending 
only  on  x  and  the  square  root  of  a  polynomial  of  not  higher 
than  the  second  degree  in  x  can  be  integrated,  and  the  result 
expressed  in  terms  of  known  functions. 

EXERCISES  ON  CHAPTER   IV 
^     r  (-x^  +  ^x)dx  2.    r    \ix-a)^-\'\dx 

'  J  (x2  +  2)2Vx^Zri  '  J  2(x  -  a)t  -  (x  -  a)i 

[Substitute  >/i^^n:  =  z.i  3.  r.l£i+^Lzil^. 

•'(x2+l)2(xa+2)i 


139-140.]^ 

4.    C ^ 

•^  X  +  y/x  —  1 


EC^MATION  BY  RATIONALIZATION  237 

dx 


dx 


x  +  Vx^-1 


-J 


dx 


8. 


(a  +  a:)^ 

(2  -  3  a:-^)  rfa: 


J 


a;  —  3  376  +  5  a;i! 


v^ 


'•I 


Vx2  -  lVv^TT  +  Va:-l 


10 


[Assume  Vx  +  1  +  Va;  —  1  =  2.] 
of  a: 


J(a;2+a2) 


Va;2  -  a2 
[A  ssume  x  =  a  sec  ^.] 


11. 


1+V^ 


Vx 


ri  +  v 

Jl  +  v 


^a:. 


12. 


(/x 


+  x  (1  +  a;)2 

I  Substitute  ^-^  =  ^8.1 
L  1  +  a;  J 


CHAPTER  V 

INTEGRATION    OF    TRIGONOMETRIC    AND     OTHER     TRAN- 
SCENDENTAL FUNCTIONS 

141.  In  regard  to  the  integration  of  trigonometric  func- 
tions, it  is  to  be  remarked  in  the  first  place  that  every 
rational  trigonometric  function  can  be  rationally  expressed 
in  terms  of  sine  and  cosine. 

It  is  accordingly  evident  that  such  functions  can  be  inte- 
grated by  means  of  the  substitution 

sin  x=-  2. 

After  the  substitution  has  been  effected,  the  integrand 
may  involve  the  irrationality 

Vl  —  252(=  cos  a;). 

This  can  be  removed  by  rationalization,  as  explained  in  the 
preceding  chapter,  or  the  method  of  reduction  may  be 
employed. 

The  substitution  cos  a;  =  2  will  serve  equally  well. 

It  is  usually  easier,  however,  to  integrate  the  trigonometric 
forms  without  any  such  previous  transformation  to  algebraic 
functions.  The  following  articles  treat  of  the  cases  of  most 
frequent  occurrence. 

142.  jsec^**xdx,     j  cosec^^xdx. 

In  this  case  n  is  supposed  to  be  a  positive  integer. 
If  sec^^a?  dx  be  written  in  the  form 

Bec^'^x .  sec'a;  dx  =  (l-\-  tan'a;)"-^ (tan  a;), 


(/ 

Ch.  V.  141-143.]     TRIGONOMETBIC  FUNCTIONS  239 

the  first  integral  becomes 

C(i2iYi^x  +  l)«-i<:7(tan  x). 

If  (tanV  +  1)""^  be  expanded  by  the  binomial  formula 
and  integrated  term  by  term,  the  required  result  is  readily 
obtained. 

In  like  manner, 

J  cosec^"2J  dx=  \  cosec^"~^a;  •  cosec^a;  dx 

=  -C^cot^x  +  l)"-^^(cotrc). 

This  last  form  can  be  integrated,  as  in  the  preceding  case,  ' 

by  expanding  the  binomial  in  the  integrand.  (^  < 

The  same  method  will  evidently  apply  to  integrals  of  thei  ^  j 

foim  \    f- 

rtan"*a;sec^«a;cZrr,  Jcot"*a;  cosec^^o:  6^ic,  V^/V 

in  which  m  is  any  number.  \^  X  "^^ 


dx  K    rn-cosxW^ 


cof*a: 


^•1 

2.  J  cosec^a;  dx,  "^   }  ^^^^^  ^^^^^  (cos4^  _  sin^a:)^'"/^^ 

3.  (sec^xdx,  '  ^'   f  .  ^"^       (=(tan-^xsec^xdx).  f 
J  J  sin^a; cos x      -^  <\ 

Jdx  g    rcos%  dx  XJ^^^^ 

143.    fsec'**ictan2»*  +  iicc?ic,  J  cosec"*a?cot2'»  +  ia5^iC. 

In  these  integrands  n  is  a  positive  integer,  or  zero,  so  that    ^ 
2  w  +  1  is  any  positive  odd  integer,  while  m  is  unrestricted./--  >w 


240  INTEGRAL   CALCULUS  [Ch.  V. 

The  first  integral  may  be  written  in  the  form 

J  sec"*"^a;  tan^"a;  •  sec  x  tan  x  dx 

=  j  sec"*"^a;  (sec^a;  —  V)'*d  (sec  a:), 

which  can  be  integrated  after  expanding  (sec^ic  —  l)**  by  the 
binomial  formula. 
Similarly, 

J  cosec"*a;  cot^"^^a;  dx  =  j  cosec^""^a;  cot^"a;  •  cosec  x  cot  x  dx 

=  —  I  cosec"*"^a;  (cosec^a;  —  l)"c?(cosec2;). 


EXERCISES 


f^ 


1.  j  sec%  tan^a;  dx.  5.    \  tan^x  dx. 

2.  fcosecSarcotSarda;.  6.    (^^^^^^^=(sec''-^xta.n^xdx\ 
J  J     cos'*x       •' 

o     fsec  ax  -.^  ^     c , 

4.  J  sin  X  Go\?x  dx.  8.  J  cot  x  dx. 

y       144.    (tun^^ocdx,     KcoV^xdx, 

The  first  integral  can  be  treated  thus : 

j  tan**  a?  dx=  j  tan""^  •  tan^a;  dx 

= J  tan"-2  a;  (sec^  a;  —  1 )  ia; 

=  ^.^^-Ct^n^-^xdx. 

When  w  is  a  positive  integer,  the  exponent  of  tan  x  may 
be  diminished  by  successive  applications  of  this  formula 
until  it  becomes  zero  (when  n  is  even),  or  one  (when  n 
is  odd). 


143-144]  TRIGONOMETRIC  FUNCTIONS  241 

In  like  manner, 

J  cot"  a?  dx=\  GoV^-^x  Qoi^x  dx 

s=  j  cot'*~^a;(cosec2a;—  V)dx 

=  —  — — I  GoV'^x  dx. 

n  —  1      *^ 

Since  tana;  and  cot  a;  are  reciprocals  of  each  other,  the 
above  method  is  sufficient  to  integrate  any  integer  power 
of  tan  a;,  or  cot  x. 

Another  method  of  procedure  would  be  to  make  the 
substitution  tan  x  =  z,  whence 

2"  dz 


ft3in''xdx=f^ 


P 
If  the  exponent  w  is  a  fraction,  say  n  =  —^  the  last  integral 

can  be  rationalized -by  the  substitution  z  =  u^. 

It  is  evident  from  this  that  any  rational  power  of  tangent 
or  cotangent  can  be  integrated. 

EXERCISES 

1.  J  cot*xdx. 

2.  \ta.ii^(ixdx. 

3.  J  (tan  X  —  cot  xy  dx, 

4.  j'(taii«ar  + tan«-2a;)da;. 

5.  I  tan^  X  dx. 

When  w  is  a  positive  integer  show  that^ 

6.  f  tan^n  X  dx  =  ^^^^^^  -  *^B!!i!5  +  •.•+(-  l)«-i  (tan  x  -  x). 
J  2n-l        2n-3  v        /       v  / 

7.  (t^n^n+i^dx  =  *^^^  -  ^-^^+ ...  +  (- l)~-Kitan2a:+logcos:r). 


242  INTEGRAL   CALCULUS  [Ch.  V. 

145.  Tsinw*  X  COS"  x  dx, 

(a)  Either  m  or  n  a.  positive  odd  integer. 
If  one  of  the  exponents,  for  example  m,  is  a  positive  odd 
integer,  the  given  integral  may  be  written 

gijjw-i  ^  (3Qgn  ^  gjjj  xdx=  —  j  (1  —  cos^  x)  ^  cos"  a:c?(cos  x). 

Since  m  is  odd,  w  —  1  is  even,  and  therefore  ^  ~     is  a 

positive  integer.  Hence  the  binomial  can  be  expanded  into 
a  finite  number  of  terms,  and  thus  the  integration  can  be 
easily  completed. 

Ex.  1.    j  sin^a:Vcosarc?a:. 

According  to  the  method  just  indicated  this  integral  can  be  reduced  to 
—  J  sin*  a;  Vcos  x  d(cos  x)  =  —  j  (1  —  cos2a;)2'(cosa;)^<f(cosa:) 
=  —  I  cos* a:  +  ^  cos2  X  -  ^cos  i*  a;. 
Ex.  2.  (sin^xdx.  Ex.  5.  f    sin^xrfx 


"Vcos  a: 

Ex.3,  i  sin^  a:  cos*  a;  dx.  ^     •  a    ^ 

•^  Ex.6.  |_^1L£££_. 

r^^cB^  ^^  —  cos  a; 

Ex.4.  (^2t£dx. 
J  sin  a: 

\  Y^     Q>)  w  +  ^  an  even  negative  integer. 

In  this  case  the  integral  may  be  put  in  the  form 

/sin"*  X  C 
CQgm+»  xdx=  \  tan™  x  sec'^™"^"^  x  dx^ 
cos"*  X                         ^ 

which  can  be  integrated  by  Art.   142,  since  the  exponent 
—  (w  +  w)  of  sec  a;  is  an  even  positive  integer. 


145.]  TRIGONOMETRIC  FUNCTIONS  243 

Ex.7,    f^^rfa:. 

COS^  X 

The  integration  is  effected  in  the  following  steps : 

^  Vcosa;  cos^  x 

=  I  tan^ x  (tan2 x  +  l)d (tan  a;) 

=  2tanta:(^+ ^tan2a;). 

Ex.    8.    i^-^^dx.  Ex.11,    f  .  /^        . 

-^  sin*  a;  •^  sin*  a;  cos^  x 

Ex.    9.    f-;^.  Ex.12,    r 


sin^a;  -^  Vsin^  a;  cos^  x 

Ex.10.    f5!:!?l^dx.  Ex.13.    fHH^dx. 

-^  sni^a:  -^  cos^+^a; 

(c)  Multiple  angles. 

When  m  and  n  are  both  even  positive  integers,  integration 
may  be  effected  by  the  use  of  multiple  angles.  The  trigo- 
nometric formulas  used  for  this  purpose  are 


.2^  _ 


sma;  cos  a;  = 
Ex.  14.    j  sin^a;  cos*  a:  c?a:. 


1  —  cos  2  X 
sin^  X  = ) 

1  -h  COS  2  X 

2 ' 

sin  2  a; 


fsin^  X  cos*  xdx=  \  (sin  a:  cos  xY  cos^  x  dx 

_  rsin2  2a:  1  +  cos  2  a;  . 
~~J       4  2  ^ 

=  i  f  sin2  2xdx+  t^  f  sin2  2  a;  cos  2  a;  c?  (2  a:) 
1  ri  —  cos  4 a:  ,^  ,    i   sin^  2  x 

=  -jJ^  a;  —  ^  sin  4  a:  +  j^  sin^  2  a:. 


244  INTEGRAL   CALCULUS  [Ch.  V. 

Ex.15.    J  cos*  ar  sin 2  a;  rfar.  Ex.17,    j  sin*  a;  cos*  arrfx. 

Ex.16,    i  silica:  cos®  a: rfa;.  Ex.18.    J  (sin* a;  —  cos* x)*c?a;. 


Integrate  the  two  following  by  the  aid  of  multiple  angles. 


Ex 


.19-    f^ 

J  sin* 


dx 


<l 


sin*  X  cos*  X    * 


Ex.  20. 


J  sma: 


dx 


cos"x—  sin"  a:  cos  x 


Integrate  the  following  by  any  of  the  preceding  methods. 

Ex.  21.    r^inlf  dx  =  r(l  -  cos^a:)* rf^r  =  f  (sec^ x-2  +  cos^ a:) dx, 
^cos^a;  ^        cos^a;  ^^  ^ 

Ex.24.    (xW^^^dx. 
Ex.  25.    r 


cos^ 

Ex.22.    r22£!£j^. 
•^     sin*  x 


J  Ex.  23.    f^'''-^%/x. 
^        x^ 

[Substitute  a:  =  a  sin  OJ] 
doc 


dx 


xWx^  -  a* 
[Substitute  x  =  a  sec  0."] 


dx 


146.    f       f'       ,   f-    ,  , 

Write  a  H-  6  cos  a;  in  the  form 
a^cos2 1  +  sin2 1^  +  j^cos^  |  -  sin^  |) 


Then  f       ^^        ^  _1_  f 

•/  a  4-  ^  cos  a;     a  —  bJ  a  -\- 


sec^-  c?(  - 


Utan'l 
a  —  0  2 


VS^TTp 


tan" 


tan 


^  a  — 


146-146.]  TBIGONOMETRIC  FUNCTIONS  245 

This  result  has  a  real  form  provided  that  a  is  numerically 

greater  than  5,  since  and  a^  —  b^  are  then  positive. 

a  —  0 

When  a  is  numerically  less  than  b  the  integral  may  be 
written 


/- 


sec^  r-dl  - 
dx ^     2      /-  2    V2 

b  cos  a;      a  —  bj        2^__  ^  +  <^ 
2      5  —  a 


a; 


-1      ,  2      ^5-a 

log 


tan-  +  \T-^-- 
2       ^b  —  a 

The  integration  of 

ci?a; 


f— 


6  sin  a? 
is  effected  by  making  the  substitution 

2^  =  3^  +  2' 

which  reduces  the  integral  to 

_^ 


b  0,0^  y 
The  preceding  results  may  then  be  applied. 

EXERCISES 

2      C dx 2      r ^^ 

J  a^sin^a:  +  ft^cos^ar  *    J  a  sin  a:  +  6  cos* 

Suggestion.    Write  the  denominator  of  Ex.  2  in  the  form 


2a  sin|  cos?+  6(cos2|-  sin2^), 


divide  numerator  and  denominator  by  cos'^  ^,  and  replace  tan  r  by  a  new 
variable. 


246. 

INTEGRAL 

CALCULUS                           [Ch.V. 

3. 

r       dx 

J5  +  3cos2a; 

6.     f               ^^ 

J  (asinx  +  ficosar)^ 

4. 

J  5  -  Ssina; 

7.     f       ''^ 

J  1  +  COS^I 

5 


(/a; 


Jl-2sin2a: 

147.    \  e^  sin  nx  doc,    \  ef*^  cos  nx  dx. 

Integrate  J  e'*^  sin  nx  dx  by  parts,  assuming 

u  =  sin  wa;,  and   dv  =  e°^  c?a;. 
This  gives 

J  e'*'^  sin  nxdx  =  -  e^  sin  wa; j  g"*  cos  nx  dx,        (1) 

Integrate  the  same  expression  again,  assuming  this  time 

u  =  e"^,     dv  =  sin  nxdx. 
Then 

j  e***  sin  nxdx  — e*^  cos  nx-\--  Ae"^  cos  7ia:6?a;.       (2) 

Multiply  (1)  by  -  and  (2)  by  -  and  add.     The  integrals 
n  CL 

in  the  right  members  are  eliminated,  and  the  result  is 

r««  sin  nx  dx  =  «°"(«si»>»a;-ncns«^). 
•^  •  d?'  +  n^ 

By  subtracting  (1)  from  (2),  the  formula 

Ce-'  cos  rmdx^  """  <^  "^"  ^^  +  ^  ^^^  ^^^^ 
is  obtained. 

EXERCISES  ON  CHAPTER  V 
1.  Show  that 

(1  +  n)  J  8ec'»+2  xdx  =  tan  x  sec"  a:  +  n  J  sec  a:  dx. 
Integrate  by  parts,  taking  u  =  sec"  x,  dv  =  sec'*  x  dx. 


146-147.]  TRIGONOMETRIC  FUNCTIONS  247 

2.  Show  that 

(1  +  n)  I  cosec^+2  x  dx  =  —  cot  x  cosec«  a;  +  «n  J  cosec"  x  dx, 

J  sin  a;  COS  a;  *  J  cos^ar 

l     4     C ^^  8.    fc^cos-rfa;.     . 

[Put  a;  =  cos  ^.]  9.  p^^^^dx,    ^i 


■     5-  J"^SiJF"  "^     10.  j^'^sin" 


•sin  a:  c?a; 


xdx. 


dx 


6.    i :~;r~*  '  11     r^''  sin  2  a:  sin  a:  dx. 

Jcoszsin^a;  J 

[Suggestion.    2  sin  2  a;  sin  a;  =  cos  x  —  cos  3  a:.] 

12.  Show  that 

f  sin  aa:  sin  fta: ^a:  =  HLl^-^IL^  _  ££i^±*^ 
J  2(a-b)  2  (a  +  6) 

Use  the  trigonometric  formula 

sin  a  sin  ^  =  ^  [cos  (a  —  P)  —  cos  (a  +  ^)]. 

13.  Show  that 

f  sin  ax  cos  5a:  ^a:  =  -  "^,^  (^  "  ^>  -  "",%(^  +  ,^X 
^  2  (a  -  6)  2  (a  +  6) 

14.  Show  that 

f  cos  ax  cos  6ar  dx  =  ^^"  ^^  "  ^>/  +  ^^^  f  ^  "^  \)^. 
J  2(a-b^  2(a  +  b^ 


2(a-b)  2(a  +  b) 

.    j  sin«  a:  cos*  a;  <?a;.  17.  j  (tan  a;  +  cota:)®rfa:. 

C #       ■■■.  /•  i^a: 

•^  sin2a;cos2a;  *  J( 


(1  +  cos  a;)  8 


,^ 


v 


CHAPTER   VI 
INTEGRATION  AS  A  SUMMATION 

148.  In  the  preceding  five  chapters  various  methods  of 
integration  have  been  explained.  The  final  object  in  every 
case  has  been  to  determine  a  function  F(x)  such  that  its 
derivative  should  be  identical  with  a  given  function  f(x). 
It  is  now  proposed  to  analyze  this  idea  a  little  more  fully, 
and  to  show  that  it  readily  leads  to  a  view  of  integration 
which  is  of  the  highest  interest  and  importance. 

In  order  to  obtain  the  derivative  of  a  function  F(x)  it  is 
necessary  in  the  first  place  to  determine  the  increment 

F{x  +  ^x)-F{x)  (1) 

which  the  function  F{x)  takes  when  the  independent  vari- 
able X  takes  the  increment  Aa;. 

The  expression  (1)  can  be  put  in  a  form  more  convenient 
for  present  purposes,  if  it  be  assumed  that  for  all  values  of  x 
under  consideration  F(x  +  Aa;)  can  be  expanded  by  means  of 
Taylor's  theorem  [Art.  41,  p.  66] .     This  expansion  is 

F(x  +  Aa;) 
,     =  Fix)  +  F>(x)^x  -f  ?^  (i^xy  +  ^^^^(Ar)3  +  .... 

From  this,  by  transposing  FQx')^  the  increment  (1)  is  ob- 
tained in  the  form  of  a  series,  viz., 

F{x-\-/:^')-F(x) 
=  F(.)A.  +  A.[^A.4-^(A.y-H  ...J 

n=/(a:)Aa;4-<^(a:)Aa;.  (2) 

248 


Ch.  VI.  148.]      INTEGRATION  AS  A    SUMMATION  249 

In  the  last  expression  (/>(a;)  has  been  written  for  brevity  in 

place  of  the  series  in  brackets,  and  f^x)  is  the  equivalent  of 

F'(x)^  since  by  supposition /(a;)  is  the  derivative  of  F(x). 

Suppose  now  that  the  variable  x  starts  with  a  given  value 

a  and  increases  until  it  reaches  another  given  value  h.     The 

function  F(x)  will  change  accordingly,  beginning  with  the 

value  F(^a)  and  ending  with  F(h),     The  difference  between 

these  two,  viz., 

F(h)  -  F{a) 

can  be  determined  by  the  aid  of  (2)  in  the  following  manner. 

Let  the  variation  of  x  from  a  to  5  be  imagined  to  occur  in 

successive  steps,  first  from  a  to  a-\-  Ax^  then  from  a  +  Ax  to 

a-\-2  Axy  and  so  on.     The  increment  which  the  function  F(^x} 

takes  at  the  first  step  of  the  change  is  F(a-\- Ax^  — F(a). 

Its  value  is  found  by  giving  x  the  value  a  in  formula  (2). 

That  is, 

F(ia  +  Arr)  -  F^a}  =f(a)Ax  +  0(a)Aa;. 

The  increment  that  F(x^  takes  at  the  second  step  is 

F(a  +  2  Ax}  -  F(a  +  Ax)  =f(a  +  A2:)A2;  +  <^(«  +  Aa;)A2;, 

the  right  member  of  which  is  found  by  substituting  x=a-\-Ax 
in  (2).  In  like  manner,  by  giving  x  the  values  a-\-2Ax, 
a  +  ^Ax,  ...,  a-{-(n  —  V)Ax,  the  additional  equations  are 
found: 

F{a  +  3  A^^)  -1^(^  +  2  Ax)  =f(a  +  2  A^^)  Aa;  +  (/>(«  +  2  A:r)  Aa^, 

F(a  -{-4:  Ax)-  FQa  +  3  Ax)  ^f{a  +  3  A2;)Aa;  +  ^(a  +  3  Aa;)  Aa;, 


F{a-\-n  Ax)-'F{a->rn-\  Ax)^  fia  -\-  n-\  Ax)Ax 

•f  <^(a  +  71—1  Aa;)Aa;. 
Assume  a-\-nAx  =  'b^  (3) 


250  INTEGRAL   CALCULUS  [Ch.  VI. 

and  substitute  in  the  first  term  of  the  preceding  equation. 
The  addition  of  the  above  n  equations  then  gives 

=  Aa:[/(a)+/(a  +  A:r)+/(a  +  2Aa;)+  .••  +/(«  +  ^T^l  Arr)] 
H-Aa;[(/)(a)  +  <^(a  +  A2;)  +  <^(a  +  2A2:)H f-</)(«  +  w-l  Aa:)]. 

This  expression  for  F(h')  —  F(^a)^  while  depending  on  the 
given  function  /(a:),  contains  also  a  series  of  successive 
values  of  <j)(x)^  viz., 


A2:[(^(a) +  </)(«+ Aa^)+  .••  +(/>(«  + /i— lAa:)]. 

This  latter  can  be  gotten  rid  of  by  taking  its  limit  as  Lx 
approaches  zero. 
For,  since 

.,  .      F"(x).      ,  F'"{x^,.    .2  , 
<^(^)  =  — ^Aa:+^^^(A2:)2+  ..., 

it  follows  that 

and  hence,  if  ^  denote  the  numerically  greatest  term  of  the 

series 

<^(a)+(^(a  +  Arc)+  •-, 
then 

Ax [</)(a)  +  ^(a  +  Aa;)  +  •••]  |  <  |  Aa:[^  +  O  •••  (n  terms)] 

|<|Aa;-w<l> 
\<\(h-a)^. 
But  since,  on  account  of  (4), 

a"2o*=o. 

it  follows  that 
and  hence 

lim 


=  Ax'i  0  [/(«)+/(«  +  ^^)  +•••+/(<»  +  »-  1  Ax)]A2:.    (5) 


148.]  INTEGRATION  AS  A   SUMMATION  251 

The  second  member  of  (5)  is  denoted  for  brevity  by  the 
symbol 

and  is  called  the  definite  integral  of  fQc)  between  the  limits 
a  and  h. 

Suppose  one  of  the  limits,  say  the  upper  limit  5,  is 
regarded  as  variable,  while  the  other  has  a  fixed  value. 
To  emphasize  this  assumption  concerning  the  variability  of 
5,  let  it  be  replaced  by  the  letter  x.  Then  equation  (5) 
may  be  written 
lim 


•^(*)  =  Ail  0  [/(«)  +/(«>  +  Aa;)  +  -  +/(a  +  n-16.x)-\^x 

+  Fia).  (6) 

Here  the  term  FQa)  has  a  fixed,  although  arbitrary,  value 
depending  on  the  particular  choice  that  is  made  for  the  con- 
stant a.     It  may  be  regarded  as  a  constant  of  integration. 

Formula  (6)  expresses  in  two  steps  the  solution  of  the 
problem  of  determining  the  function  F(x') : 

(1)  Find  the  sum  of  the  series  of  n  terms 

fia),  f{a  +  A:r),  f(a  +  2  Ao;),  .-.,  /(a  -V{n-  l)A:i:), 

these  being  the  values  of  the  given  function  f{x)  corresponding 
to  the  n  equidistant  values  of  x^ 

a,  a  +  Arr,  a  +  2  Ax,  •••,  a -{-  (n  —  1)  Aa;. 

(2)  Find  the  limit  of  the  product  of  this  sum  hy  Ax,  as  Ax 
approaches  zero  while  n  increases  to  infinity,  subject  to  the  con- 
dition nAx  =  X  —  a. 

The  addition  of  an  arbitrary  constant  of  integration  makes 
the  solution  the  most  general  possible. 

The  method  just  formulated  for  determining  the  integral 
F(x)  of  a  given  function  f(x)  is  not  suitable  for  the  actual 


252  INTEGRAL   CALCULUS  [Ch.  VI. 

work  of  integration,  since,  with  few  exceptions  (cf.  Exs.  1, 
2  below),  the  summation  of  the  series  in  the  right-hand 
member  of  (6)  presents  insuperable  difficulties. 

On  the  other  hand,  formula  (5)  admits  of  a  very  simple 
geometrical  or  physical  interpretation  in  most  of  the  applica- 
tions of  the  calculus,  and  herein  lies  one  of  its  chief  merits. 
It  places  before  one  a  very  convenient  and  useful  formulation 
of  many  of  the  problems  of  geometry,  mechanics,  physics, 
etc.,  the  final  solution  of  which  is  most  readily  effected  by 
the  evaluation  of  the  definite  integral 

'f(x)dx 

in  the  following  manner.  First  obtain  the  function  F(x) 
by  integrating  f{x)dx  according  to  the  methods  already 
explained  in  the  preceding  chapters. 

Determine  T'Q))  and  JP(a)  by  substituting  the  limits  h  and 
a  in  the  result.  Finally  subtract  ^(a)  from  FQ)),  This 
gives 

£f(x)dx  =  F(h^  -  Fii^oL) 

as  the  value  of  the  definite  integral. 

Ex.  1.   Given /(x)  =  e*,  find  F{x)  by  the  method  of  summation. 
For  the  sake  of  brevity  write  Ax  =  A.     Then  formula  (6)  gives 

^(^)  =  ^^^Q  [e"  +  c*+*  +  e*-^"  +  -  +  e«+(«-i>*]A  +  F(a). 
The  sum  in  the  right  member  may  be  written 

gan  ^.  g»  4.  g8»  4.  ...  4.  c(H-i)»]A  =  e«l^=^  •  h 

1  —  e* 
(by  the  formula  for  summing  a  geometric  series) 


1  —  c* 

=  <-(^--i).-jf^- 


148-149.] 


INTEGRATION  A 8  A   SUMMATION 


253 


As  h  is  made  to  approach  zero  the  factor  becomes  indetermi- 

e*—  1 
nate.    Its  limit  is  found  by  the  method  of  Chapter  V  (p.  77)  to  be 

lim       ^ 


4 


1. 


A  =  0  e»  _  1 
Hence  ^^^^  e«  [1  +  e*  +  . . .  +  e(«-i)»]A  =  c«  _  e«, 

and  accordingly  F(x)  =  \  e'^dx  z=  e'^  —  e** -\-  F(a)  =  e*  -j-  C, 

in  which  C{=  F(a)  —  e«)  may  be  regarded  as  an  arbitrary  constant  of 
integration. 

Ex.  2.   Given /(a:)  =  ax,  find  \  axdx  by  the  method  of  summation. 

149.  Geometrical  interpretation  of  the  definite  integral  as  an 
area.  Let  the  values  of  the  function  f(x)  be  represented  by 
the  ordinates  to  a  curve.     Its  equation  would  then  be 

It  is  proposed  to  find  an  expression  for  the  area  bounded  by 
this  curve,  the  a;-axis, 
and  two  ordinates  AP 
and  BQ^  correspond- 
ing to  two  given 
values  oi  x^  x  =  a  and 
a;  =  5,  respectively. 

Let  the  interval 
from  J.  to  5  be 
divided  into  n  equal 
intervals  AA^^  ^i-^g, 
•  •  •,  An-\B  each  of 
magnitude  Aa;,  so  that 

interval  AB  =  h  —  a  =  nAx. 
At  each  of  the  points  of  division  A,  A^,  -  -  -,  B  erect  ordi- 
nates, and  suppose  that  these  meet  the  curve  in  the  points  P, 
Pj,   •  •  •,    Q,     Through  the  latter  points  draw  lines  PB^^ 
PiB^,  •  •  •,  P„_i-B^  parallel  to  the  a;-axis. 


A,  A,  A^^B 

FiQ.  61. 


254  INTEGRAL   CALCULUS  [Ch.  VI. 

A  series  of  rectangles  PA-^^  ^1^2.^  •  •  •  is  thus  formed,  each 
of  which  lies  entirely  within  the  given  area.  These  will  be 
referred  to  as  the  interior  rectangles.  By  producing  the 
lines  already  drawn,  a  series  of  rectangles  SA-^<,  ^1^2'  •  •  •  is 
formed  which  will  be  called  the  exterior  rectangles.  It  is 
clear  that  the  value  of  the  given  area  will  always  lie  between 
the  sum  of  the  interior,  and  the  sum  of  the  exterior  rec- 
tangles, or,  expressed  in  a  formula, 

PA^  +  P^A^  +  ...  +  Pn-iB  <  area  APQB  <  SA^  +  S^A^ 
+  ...  +  S^_,B.  ^  (7) 

The  difference  between  the  sum  of  the  exterior  and  the 
sum  of  the  interior  rectangles  is 

SR^  +  ^A  +  -  +  ^n-iBn  =  rectangle  ^^1^=  TQ  .  ^x. 
If  the  function /(a;)  does  not  become  infinite  as  x  varies  from 
a  to  5,  TQ  will  be  finite  and  hence  TQ  •  ^x  will  approach  zero 
simultaneously  with  Aa;.  Hence  the  limit  of  the  sum  of  the 
exterior  rectangles  equals  the  limit  of  the  sum  of  the  inte- 
rior rectangles.  From  (7)  it  follows  that  the  area  is  equal 
to  the  common  limit  of  these  two  sums. 

To  determine  this  sum  observe  that 

Rectangle  APR^A^  =  AP  -  AA^  =/(a)  •  Ax. 
Similarly  A^P^R^A^  =/(«  +  Aa;)  •  Ax, 

A„_,P„_,R„B  =fia  +  ir=l.  Ax)  .  Ax, 
Adding, 

sum  of  rectangles 

=  [/(«)  +/(«  +  A^)  +  -  +/(«  +  ^ir^Ax')-]Ax, 
Hence,  by  requiring  Ax  to  approach  zero, 
Area  APQB 
=  Ai'So[/(«)  +/(«  +  ^^)  +  -  +/(''  +  ^^^^^  Aa;)]Aa:.     (8) 


149-150.] 


INTEGRATION  AS  A   SUMMATION 


255 


The  expression  just  obtained  for  the  area  is  identical  with 
that  occurring  in  the  right-hand  member  of  (5),  and  affords 
one  of  the  simplest  and  most  interesting  of  the  geometrical 
interpretations  of  that  formula.     Thus 


Xb  r*h 

f(x)dx  =   I    ydo, 


(9) 


150.  Generalization  of  the  area  formula.  Positive  and 
negative  area.  Instead  of  taking  the  limit  of  the  sum  of 
the  interior  (or  exterior)  rectangles,  a  more  general  pro- 
cedure would  be  to  take  a  series  of  intermediate  rectangles. 


Fig.  62. 


Let  x-^  be  any  value  of  x  between  a  and  a  +  Aa;,  x^  any  value 
between  a  4-  Aa;  and  a  +  2  Lx,  etc.  Then  f{x^Lx  would 
be  the  area  of  a  rectangle  KLA^A  (Fig.  62)  intermediate 
between  PA^  and  SA^ ;  that  is, 


Likewise 


PA^<f{x^Lx<SA^.  , 


256  INTEOUAL   CALCULUS  [Ch.  VI. 

Hence 

Sum  of  interior  rectangles  <  [/(i^{)+f(^2)-\ — ]  Aa; 

<  sum  of  exterior  rectangles, 
and  therefore  (cf.  Fig.  61), 

Area  APQB  =  ^^  0  [/(^i)  +/C^2)  +  -  +/(^n)] Aa:.    (10) 

If  the  area  to  be  found  is  entirely  above  the  a^-axis,  the 
ordinates  are  all  positive.  If  at  the  same  time  Ax  be  taken 
positive  (that  is,  if  6  >  a),  formula  (8)  or  (10)  gives  a  posi- 
tive sign  to  the  area.  On  the  other  hand,  the  area  is  nega- 
tive if  below  the  a;-axis. 

If  the  curve  i/  =f{x)  is  partly  above  and  partly  below  the 
jc-axis,  the  value  of  the  definite  integral  (8)  will  be  repre- 
sented by  the  algebraic  sum  of  the  positive  and  negative 
areas  limited  by  this  curve. 

151.  Certain  properties  of  definite  integrals.  From  the 
definition  of  the  definite  integral  I  f(x)  dx  as  the  limit  of  a 
particular  sum  [formula  (5),  p.  250],  certain  important 
properties  may  be  deduced. 

(a)  Interchanging  the  limits  a  and  b  changes  the  sign  of  the 
definite  integral. 

For  if  X  starts  at  the  upper  limit  h  and  diminishes  by  the 
addition  of  successive  negative  increments  (  —  Aa;),  a  change 
of  sign  will  occur  in  formula  (5),  giving 

F(ia^-Fih)  =  Jj(ix)dx. 

Hence  j^f^^^  ^^  =  "X'^^^^  ^^'  ^^^^ 

(3)  If  c  he  a  number  between  a  and  b  (a<c<b'),  then 

j^V(2:)  dx  =£fix)  dx  +  J /(a:)  dx,  (12) 


160-152.] 


INTEGBATION  AS  A   SUMMATION 


257 


((?)   The  Mean  Value  Theorem. 

The  area  APQB  (Fig.  63),  which  represents  the  numerical 
value  of  the  definite  integral  may  be  determined  as  follows : 

Let  an  ordinate  JfiVbe  drawn 
in  such  a  position  that 

area  PSN  =  area  NB  Q. 

If  f  denote  the  value  of  x  cor- 
responding to  the  point  iV,  then 
illfiV^  =  /(f),  and 

Area  APQB  =  rectangle  ASBB 

=  M]Sr'AB=fQ)(b-ay 

Hence, 

jr>(;r)cfo=/(?)(6-«),        (13) 

in  which  f  is  some  value  of  x  between  a  and  5.     This  result 
is  known  as  the  Mean  Value  Theorem.     (Compare  Art.  45.) 
The  theorem  may  be  expressed  in  words  as  follows : 

The  value  of  the  definite  integral 


^Jix)dx 


is  equal  to  the  product  of  the  difference  between  the  limits  by 
the  value  of  the  function  f(x)  corresponding  to  a  certain  value 
x=  ^  between  the  limits  of  integration. 

152.  Definition  of  the  definite  integral  when  /(a?)  becomes 
infinite.  Infinite  limits.  In  the  preceding  sections  it  has 
been  assumed  that  /(a?)  is  always  finite  so  long  as  x  remains 
within  the  prescribed  limits.  It  is  now  necessary  to  examine 
the  cases  in  which /(a;)  is  infinite. 

Suppose,  in  the  first  place,  that  f(x^  becomes  infinite  at 
the  upper  limit  x=b,  but  is  elsewhere  finite.     In  that  event, 


258  INTEGRAL   CALCULUS  [Ch.  VI. 

take  for  upper  limit  a  value  x  —  x\  which  is  less  than  5, 
a<x'<h.     Then,  according  to  the  preceding  results, 

F(x')  -  F(a)  ^^J(x)dx, 

Now  let  x^  increase  and  approach  h  as  limit.  If  at  the 
same  time  the  integral 

£'f(x)dx  (14) 

approaches  a  definite,  finite  limit,  that  limit  will  be  defined 
as  the  value  of  the  integral 

in  the  case  under  consideration  ;  that  is, 

jj(x-)dx  =  )%jj(x)d^.  a/<b. 

On  the  other  hand  the  integral  (14)  may  increase  without 
limit.  When  that  happens,  the  integral  will  be  said  to  have 
an  infinite  value,  or 

j  f(x)dx  =  QO. 

In  a  similar  manner,  if  f(a)  =  oo,  the  value  of  j  f{x)dx 
will  be  defined  to  be  the  limit  of  the  integral 

Cf{x)dx  a<x^<h 

as  a/  diminishes  and  approaches  a  as  limit. 

Finally,  if /(<?)  =  oo,  where  c  is  any  number  between  a  and 
6,  it  is  necessary  to  determine  the  meaning  of 

Cy(x)dx,  and    i  f(x)dx 

by  the  method  just  suggested,  and  then  add  the  two  results 
in  accordance  with  formula  (12). 


162.]  INTEGRATION  AS  A   SUMMATION  259 

Heretofore  the  limits  a,  h  have  been  assumed  to  be  finite. 
The  case  in  which  one  of  the  limits,  say  6,  is  infinite,  is 
readily  disposed  of  by  integrating  from  a  to  a  finite  upper 
limit  x\  and  then  considering  the  limit  which  the  integral 
approaches  as  x'  increases  to  infinity.  This  limit,  when  one 
exists,  will  be  defined  as  the  value  of  the  integral,  so  that       kC 

An  exactly  similar  mode  of  procedure  is  to  be  followed  if    /  j  « 
the  lower  limit  a  is  —  qo,  or  if  both  limits  are  infinite.  V 


EXERCISES 
Ex.  1.  Prove,  without  performing  the  integration,  ttiat 

J-a^  +  x'         '   '    ji 
Ex.  2.  Without  integrating  show  that 

C^    xdx    ^  r^^    xdx 

Ex.3.  If  2/ =  ^(a:)  and  y  =  if/(x)  are  the  equations  of  two  curves 
which  are  continuous  between  x  =  a  and  x  =  b,  and  such  that  to  each 
value  of  X  (a<a;<6)  corresponds  but  one  value  of  y,  prove  that  the 
area  bounded  by  these  curves  and  the  two  ordinates  x  =  a,  x  =  b  is 
numerically  equal  to 

il/(x)'\dx. 


s/ 


Clcl>(x) 

•fa, 


Ex.  4.  Prove  that  the  area  of  the  circle  {x  -  hy  ■\-  {%f  —  kY  —  r^  is 

equal  to  /^A+r      , _ 

^  *       2\/r2-  (x-hydx. 


Jn-r 

Ex. 

5. 

Evaluate 

♦'o  < 

dx 
32  4.  a;2* 

F.TT 

6. 

Evaluate 

dx 

Jx-1 

Ex. 

7. 

Evaluate 

V- 

dx 

0  (x  -  l)t 


CHAPTER  VII 

GEOMETRICAL  APPLICATIONS 

153.  Areas.  Rectangular  coordinates.  It  was  shown  in 
Art.  149  that  the  area  bounded  by  the  curve  y  =/(a;),  the 
a:-axis,  and  the  two  ordinates  a;  =  a,  a:  =  5,  is  represented  by 
the  definite  integral 

jj(x)dx^j^ydx.  ■  (1) 

In  an  exactly  similar  manner  it  can  be  shown  that  the  area 
limited  by  the  curve,  the  y-axis,  and  the  two  abscissas  «/  =  «, 
«/  =  yS,  is  represented  by 

xdy,  (2) 


£• 


It  was  remarked  at  the  end  of  Art.  150  that  when  h  is 
greater  than  a  the  integral  (1)  gives  a  positive  or  negative 
result  according  as  the  area  is  above  or  below  the  a;-axis. 

Similarly,  if  /S>a,  the  integral  (2)  gives  a  positive  or 
negative  result  according  as  the  area  which  it  represents  is 
to  the  right  or  left  of  the  y-axis. 

Whenever  it  is  required  to  determine  the  area  of  a  figure 
which  is  partly  on  one  side  and  partly  on  the  other  side  of 
the  coordinate  axis,  it  is  necessary  to  calculate  the  positive 
and  the  negative  areas  separately  and  add  the  results,  each 
taken  with  a  positive  sign.     [Cf.  Ex.  5,  p.  262.] 

154.  Second  method.  Another  method  of  determining 
the  area  is  based  on  the  result  of  Art.  10,  p.  23.  It 
was  there  shown  that   if  z   represents  the  area  measured 

260 


153-154.] 


GEOMETRICAL  APPLICATIONS 


261 


from  a  fixed  ordinate  AP  (at  a;  =  «)  up  to  an  ordinate  MB' 
corresponding   to   a   variable   abscissa  x,  then   the  deriva- 
tive of  area  with  respect  to 
X  is  equal  to  the  function 
f(x) ;  that  is 

or,  in  the  differential  nota- 
tion, 

dz  =  i/dx  =f(x)dx,' 

The  area  z  may  accordingly 
be  found  by  integrating/(a;). 


Hence   z  =  \f(x)dx  +  C. 


Fio.64. 


The  value  of  the  constant  of  integration  Q  is  determined 
by  the  condition  that  when  x  =  a^  z  must  be  zero,  since  in 
that  event  the  ordinate  MJSf  coincides  with  the  initial 
position. 

Ex.  Find  the  area  bounded  by  the  curve  y  =  log  x,  the  ar-axis,  and 
the  two  ordinates  a;  =  2,  a;  =  3. 

Axesi  APNM=  (log  xdx-\-  C 

=  x(\ogx-l)-\-C. 

Since  the  area  is  zero  when 
a;  =  2,  it  follows  that 

0  =  2  log2-2  +  C, 
■  whence 

C  =  2-2  1og2. 
Accordingly 

X  (log  a;  -  1)  +  2  -  2  log  2 
represents  the  area  measured 
from  the  ordinate  a:  =  2  up 
to  the  variable  ordinate  MN. 
When  ar  =  3  the  required  area 
1)  +  2  -  2  log  2  =  log  -2;^  -  1. 


Fig.  65. 


is  found  to  be  3  (log  3 


262 


INTEGRAL   CALCULUS 


[Ch.  VII. 


EXERCISES 

1.  Find  the  area  bounded  by  the  parabola  y  =  4  ax%  the  a:-axisj 
and  the  ordinate  x  =  h. 

2.  Find  the.  area  of  the  triangle  formed  by  the  line  -  +  |=  2  and 
the  coordinate  axes. 

3.  Find  the  area  between  the   a:-axis  and  one  semi-undulation  of 
the  curve  y  =  sin  x. 

4.  Find  the  area  bounded  by  the  semi-cubical  parabola  y^  =  ax^  and 
the  line  x  =  5. 

5.  Find  the  area  between  the  curve  y  —  sin^  x  cos  x  and  the  a;-axis, 
from  the  origin  to  the  point  at  which  a;  =  2  tt. 


Fig.  66. 


An  examination  of  the  curve  will  show  that  the  area  is  partly  above 
and  partly  below  the  z-axis.    The  curve  crosses  the  axis  at  x  =  -,  and 

at  a:  = 

2 

The  first  portion  of  area,  which  is  positive,  is  obtained  by  integrat- 
ing from  0  to  ^.     The  result  is  \.     The  next  two  portions  of  area  are 

negative,  and  are  calculated  by  integrating  from  -  to  — ^.    The  result  ia 


3^ 


|.     The  last  portion,  which  is  positive,  is  found,  by  integrating  from 
to  2 TT,  to  be  \.     Hence  total  area  =  |4-f  +  |  =  f. 


6.  Find  the  area  between  the  a;-axis  and  the  curve  y  —  a  sin  4  x, 
from  the  origin  to  x  =  tt. 

7.  Find  the  area  bounded  by  the  cubical  parabola  y  =  x*,  the  ^-axis, 
and  the  line  ^  =  8. 


154-165.]  GEOMETBICAL  APPLICATIONS  263 

8.  Find  the   area  bounded  by  the  parabola  y  =  x^  and  the  line 
y  =  x.     [Cf .  Ex.  3,  p.  259.] 

9.  Find  the  area  bounded  by  the  parabola  y  =  x^  and  the  two  lines 
y  =  X,  and  y  —  2  x. 

10.  Find  the  area  bounded  by  the  parabola  y"^  =  4:px  and  the  line 
x=  a,  and  show  that  it  is  two  thirds  the  area  of  the  circumscribing 
rectangle. 

What  is  the  area  bounded  by  the  curve  and  its  latus  rectum? 

11.  Find  the  area  of  the  circle  x^  +  y"^  +  2  ax  =  0. 

12.  Find    the    area    bounded    by  the    coordinate    axes,  the  witch 

y  =  — ,  and  the  ordinate  x  =  Xy     By  increasing  x^  without  limit, 

find  the  area  between  the  curve  and  the  ar-axis. 

13.  Find  the  area  of  the  ellipse  ^  +  ^L  =  i. 

'14.   Find  the  area  of  the  hypocycloid  x'^  +  ys  z=  a*. 

15.  Find  the  area  bounded  by  the  logarithmic  curve  y  =  a*,  the 
a:-axis,  and  the  two  ordinates  x  =  x^  x  =  x^.  Show  that  the  result  is 
proportional  to  the  difference  between  the  ordinates. 


.  Precautions  to  be  observed  in  evaluating  definite 
integrals.  The  two  methods  just  given  for  determining 
plane  areas  are  essentially  alike  in  the  processes  required, 
namely : 

(1)  to  find  the  integral  of  the  given  function  f{x)  ; 

(2)  to  substitute  for  x  the  two  limiting  values  a  and  5, 
and  subtract  the  first  result  from  the  second. 

Erroneous  results  may  be  reached,  however,  by  an  in- 
cautious application  of  this  process. 

In  practical  problems,  the  case  requiring  special  care  is 
that  in  which  f(^x)  becomes  infinite  for  some  value  of  x 
between  a  and  h.  When  that  happens,  a  special  investiga- 
tion must  be  made  after  the  manner  of  Art.  152. 


264 


INTEGRAL   CALCULUS 


[Ch.  vn. 


Ex.  1.   Find  the  area  bounded  by  the  curve  y(x-  1)^  =  c,  the  coordi- 
nate axes,  and  the  ordinate  a;  =  2. 

A  direct  application  of  the  formula  gives 

C^    cdx  c    ~|2        ^ 

area  =  \  — ^-^ —  = —     =  —  2  c, 

Joix-iy         x-Uo 

]h 
is  a  sign  of  substitution,  indicating  that  the  values 
a 

ft,  a  are  to  be  inserted  for  x  in  the  expression  immediately  preceding  the 
sign,  and  the  second  result  subtracted  from  the  first. 

This  result  is  incorrect.     A  glance  at  the  equation  of  the  curve  shows 

that/(a;)j  ==—£——     becomes  infinite  for  x  =  \.     It  is  accordingly 


£C=2 


Fia.  67. 

necessary  to  find  the  area  OCPA  (Fig.  67)  bounded  by  an  ordinate  AP 
corresponding  to  a  value  x  =  x'  which  is  less  than  1.  For  this  portion 
the  area  f(x)  is  finite  and  positive,  and  formula  (1)  can  be  immediately 
applied,  with  the  result 


area 


Jo(x-iy         (x-l)Jo         x'-l 


If  now  x'  be  made  to  increase  and  approach  1  as  a  limit,  the  value  of 
the  expression  for  the  area  will  increase  without  limit. 

A  like  result  is  obtained  for  the  area  included  between  the  ordinates 
X  =  1  and  X  =  2.     Hence  the  required  area  is  infinite. 


Ex.2.   Find  the  area  limited  by  the  curve  y^  (x^  -  a^y  =  8  x*,  the 
coordinate  axes,  and  the  ordinate  a;  =  8  a. 


165.]  GEOMETRICAL  APPLICATIONS 

2x 


265 


Since /(a:)    : 


.     becomes  infinite  for  x  =  a,it  is  necessary  in 


(a;2  -  a^)^ 
the  first  place  to  consider  the  area  OP  A  (Fig.  68)  and  determine  what 


B         X 


Fia.  68. 

limit  it  approaches  as  ^P  approaches  coincidence  with  the  ordinate 
X  =  a.     Accordingly 

area  OPA  =  C     ^^^^     =  d(x^  -  a2)il"' 

=  3(a:'2  _  a2)i  ^  3  al,  0<a/<a. 

Whence 

^'/^^[areaOP^]  =3  J.     . 
In  the  same  manner,  the  area  A'P'QB  has  the  value 

p     2xdx     ::,6at-3(a/2~a2)i,  a<a:'<3a. 

As  a:'  diminishes  towards  a,  the  area  increases  to  the  limiting  value  6  a^' 
Hence,  by  adding  the  two  results,  the  required  area  is  found  to  be 

3  at  +  6  af  =  9  ai 
The  same  result  is  found  by  a  direct  application  of  (1),  viz. : 

•^'    (a:2-a2)f  J' 

80  that  in  this  case  an  immediate  use  of  the  area-formula  gives  the  correct 
result. 


266 


INTEGRAL   CALCULUS 


[Ch.  VII. 


Ex.  3.   Find  the  area  bounded  by  curve  y  —  tan-^ar,  the  coordinate 
axes,  and  the  line  x  =  1. 

In  this  problem  we  have  to  deal  with  a 
many-valued  function  of  x.  In  fact,  to 
each  value  of  x  corresponds  an  infinite 
number  of  values  of  tan-^a:.  The  problem 
accordingly  has  an  indefiniteness  which 
must  be  removed  by  making  some  addi- 
tional assumption. 

The  cm've  y  =  tan-^  x  consists  of  an  in- 
finite number  of  branches,  corresponding 
ordinates  of  which  differ  by  integer  multi- 
ples of  TT.  Each  branch  is  continuous  for 
all  finite  values  of  x  (see  Fig*  69).  It  is 
evidently  necessary  to  select  one  of  these 
branches  for  the  boundary  of  the  proposed 
area,  and  discard  all  the  others.  Suppose,  for  example,  the  branch  ^5  is 
selected.     The  ordinate  to  this  branch  has  the  value  tt  when  x  is  zero, 


Y 

^ 

A 

B 

r^ 

0 

C 

X 

-=^^^- 

X 

-1 

Fig.  69. 


and  increases  continuously  to  tt  + 
to  1.     Hence  the  required  area  is 


as   X  increases  continuously 


f  tan-i  xdx  =  [a:  tan-^a:  -  \\og  (x^  +  1)]J 


Ex.  4.   Find  the  area  of  the  parallelogram  strip  ABCO  (Fig.  69). 


Ex.  5.   Find  the  area  between  the  cissoid  2/^ 


2u-x 


and  its  asymp- 


tote x  —  la. 

Ex.  6.  Find  the  area  inclosed  by  the  curve  xhp'  -  a^  (y^  _  x")  and  its 
asymptotes. 

Ex.  7.  Find  the  area  bounded  by  the  curve  aH  —  y{x  -  a),  the  x-axis, 
and  the  asymptote  x—a. 

Ex.  8.  Find  the  area  included  between  the  curve  (2-x)y«=a:'(a:-l)* 
and  its  asymptote. 

Ex.  9.  What  restriction  must  be  placed  on  the  exponent  k  in  order 
that  the  area  bounded  by  the  curve  (1  -  a:)*y  =  1,  its  asymptote  a;  =  1, 
and  the  coordinate  axes  may  be  finite? 

/ 


155-156.]  GEOMETRICAL  APPLICATIONS  267 

156.  Areas.     Polar  Coordinates.     Let  FQ  be  an  arc   of 

a  curve  whose  equation  in  polar  coordinates  is 

P=/W-  (3) 

Let  it  be  required  to  find  the  area  bounded  by  this  curve 
and  the  two  radii  OP  and  OQ, 

Draw  from  the  origin  a  series 
of  radii  OP^,  OF^,  .••,  OP„_i  at 
equal  angles  A^.  Let  the  coor- 
dinates of  the  points  P,  Pj,  Pg, 
...,  Q  be  («,  a),  (pi,  ^i),  (/32,  l^a), 
•  ••,  (5,  /8).  Draw  the  circle 
arcs  FB^R^,  F^R^R^,  .-.  In 
the  circular  sector  FORi^ 


radius  OF  =  a, 
arc  Pi^j  =  a  •  A^  ; 
hence  area  P  Oi^^  =  |^  a^  A^. 

Similarly  area  P^  Oi^g  =  2  Pi^  ^^» 

area  P2  07^3  =ip2^A(9, 


Fig.  70. 


area  F^_^OR„  =  |  pn-i^^O, 
The  sum  of  these  sectorial  areas  is 


(4) 


This  is  an  approximate  value  for  the  required  area  FOQ^ 
which  is  less  than  the  true  value  by  the  amounts  contained 
in  the  neglected  triangular  portions  FR^F^,  F^R^F^,  etc. 
Suppose  the  figure  FR^F^  revolved  about  0  until  it  occupies 
the  position  F'R^R^^  and  similarly  with  F^R^P^t  etc.  Then 
the  sum  of  all  the  parts  neglected  is  evidently  less  than  the 


268  INTEGRAL   CALCULUS  [Ch.  VIL 

strip  P' R^' Ii„Pn-i,  the  area  of  which  approaches  zero  as  the 
sectorial  angle  A6  is  made  to  approach  zero. 

Hence     area  P0§  =  ^^ oK«'  +  Pi"  +  P2^  +  -  +  Pn-f)^^ 


-£Wd6. 


Another  method   of   procedure    is   illustrated   in   Ex.   1 
immediately  following. 

Ex.  1.   Find  the  area  of  the  lemniscate  p^  =  a^  cos  2  0. 

Let  A  denote  the  area  of  the  sector 
POQ  measured  from  the  polar  axis  to 
an  arbitrary  radius  vector  OQ.  The  dif- 
ferential of  area  is  (Art.  88,  p.  142) 

dA  =  ip'^dO  =  ia^cos2  6  dO, 

whence,  by  integration, 

A  =^a^(cos2edB 

Fig.  71.  =^%in2^+C. 

4 

If  6  were  zero,  the  line  OQ  would  occupy  the  initial  position  OP,  and 
the  area  would  be  zero.     That  is 

^  =  0  when  ^  =  0. 

The  substitution  of  this  result  in  the  preceding  formula  gives 

0  =  0  +  C. 

Hence  C  =  0, 

and  A=^sm2e, 

4 

In  order  to  find  the  total  area  of  the  figure  put  0  =  j.    In  this  case 

OQ  will  be  tangent  to  the  lemniscate  at  0.     On  account  of  the  symmetry 
of  the  curve,  the  result  obtained  will  be  one  fourth  the  total  area,  and 

Ex.  2.   Find  the  area  of  the  cardioid  p  =  a  (1  —  cos  0).  . 

Ex.  3.   Find  the  area  of  the  three  loops  of  the  curve  p  =  a  sin  3  ft 


156-157.] 


GEOMETRICAL  APPLICATIONS 


269 


Ex.  4.  Find  the  area  bounded  by  the  hyperbolic  spiral  pO  =  a  and 
the  two  radii  p^,  p^-  Show  that  the  area  is  proportional  to  the  difference 
between  the  radii. 

Q 

Ex.  5.  Find  the  area  limited  by  the  parabola  p  =  a  sec^-  and  its 
latus  rectum.  . 

Ex.  6.   Find  the  area  of  the  circle  p  =  2a  cos  $. 
Ex.  7.   Find  the  area  of  the  four  loops  of  the  curve  p  =  a  sin  2  ^.    «/ 

Ex.  8.  ^  AB  -  H^  i^  positive,  the  equation  Ax^  ■{■  2  H  xy  +  By^=  1 
represents  an  ellipse.     By  transforming  to  polar  coordinates  find  its  area. 


kK. 


157.  Length  of  curves.     Rectangular  coordinates.     Let  it 
f       be  required  to  determine  the  length  of  a  continuous  arc  PQ 
of  a  curve  whose  equation  is  written  in  rectangular  coordi- 
nates (x^  y). 

It  is  first  necessary  to  define  what  is  meant  by  the  length 
of  a  curve.    For  this  purpose, 
suppose  a  series  of  points  P^, 
Pgi  •••'  ^n-i  taken  on  the  arc 
PQ  (Fig.  72),  and  imagine 
the   lengths   of   the   chords 
PPj,  P^P^i,^  •••to  have  been 
determined.      The   limit   of 
the  sum  of  these  chords  as 
the   length    of    each    chord 
approaches    the    limit    zero 
will  be  defined,  in   accord- 
ance with  accepted  usage,  as  the  length  of  the  arc  PQ;* 
that  is, 
arc  P  5  =  Lt  (chord  PP^ + chord  P1P2 + •  •  • + chord  P„_i  0.(5) 


Fig.  72 


♦  That  this  limit  is  always  the  same  no  matter  how  the  points  P<  are  chosen, 
so  long  as  the  curve  has  a  continuously  turning  tangent,  and  the  distances 
Pi-iPi  are  all  made  to  tend  towards  zero,  admits  of  rigorous  proof.  The  proof 
is,  however,  unsuitable  for  an  elementary  text-book.  [See  "  Rouch^  et  Com- 
berousse.  Traits  de  g^om^trie."     Paris,  1891,  part  I,  p.  189.] 


270  INTEGRAL   CALCULUS  [Ch.  VII. 

This  definition  is  immediately  convertible  into  a  formula 
suitable  for  direct  application. 

For,  let  the  points  Pj,  Pg'  "*  ^^  ^^  chosen  that 

the  lines  P-Bj,  etc.  being  drawn  parallel  to  the  a;-axis. 
Denote  by  A^  the  increment  MtPi  of  i/.  Then  the  chord 
P,_iPj  has  the  length 


It  is  clear  that  — ^  is  the  value  of  -^  corresponding  to 
Ax  dx  ^  ^ 

some  point  of  the  curve  between  P^.j  and  P,.  [Cf.  Art.  45.] 
Hence,  by  substituting  in  (5)  and  using  the  principle 
employed  in  deriving  the  area-formula  (10),  Art.  150, 


in  which  (x\  t/'}  and  (x",  y'')  are  the  coordinates  of  P  and 
Q  respectively. 

The  same  result  would  also  be  obtained  by  integrating  the 
expressions  for  the  derivative  of  arc,  (1)  and  (2),  p.  139. 

Ex.  1.    Find  the  length  of  arc  of  the  parabola  y^  =  ipx  measured 
from  the  vertex  to  one  extremity  of  the  latus  rectum. 

In  this  case  -^  =  \^, 

dx      ^x 

and  hence  length  of  arc  =  J   "Vl  +  —  rfx. 

Jo  X 

Ex.  2.   Find  the  length  of  arc  of  the  semi-cubical  parabola  ay^  =  r* 
from  the  origin  to  the  point  whose  abscissa  is  ^^ 


157-158.]  GEOMETRICAL  APPLICATIONS  271 

Ex.  3.  Find  the  entire  length  of  the  hypocycloid  xt  +  3/3  =  a^. 
Ex.  4.   Find  the  length  of  arc  of  the  circle  (x-hy  +  (y-ky  =  r\ 

X  X 

Ex.  5.   Find  the  length  of  arc  of  the  catenary  ?/  =  ^  (e*  +  e  «)  from 
the  vertex  to  the  point  {x^,  y^). 

Ex.  6.  Find  the  length  of  the  logarithmic  curve  y  =  log  x  from  a;  =  1 
to  a;  =  \/3. 

Ex.  7.  Find  the  length  of  arc  of  the  evolute  of  the  ellipse 


jAv^ 


158.  Length  of  curves.  Polar  coordinates.  When  the 
equation  of  the  curve  is  given  in  polar  coordinates,  let  the 
points  Pj,  Pg'  •**'  -^^-i  ^®  ®^  chosen  that  the  vectorial  angle  6 


Fig.  73. 


increases  by  equal  increments  A^  in  passing  from  a  point  P^ 
on  the  curve  to  the  next  succeeding  point  P^+j.  Draw  the 
lines  PiRi^i  perpendicular  to  the  radii  OPi+i.     Then 


272  INTEGRAL   CALCULUS  [Ch.  VII. 


chord  P,P,^,=  ^P,R,J  +  R,^,P,^?  [Cf.  p.  141.] 

=  VO  sin  A^)2  +  (^ + A/5  -  /o  cos  M)^ 

The  limit  of  the  sum  of  all  such  chords  will  be,  according 
to  definition,  the  length  of  the  arc  PQ.     Hence 


in  which  (/a',  ^'),  (^p"^  6")  are  the  coordinates  of  P  and  Q 
respectively. 

Ex.  1.   Find  the  length  of  arc  of  the  logarithmic  spiral  p  =  e"^  be- 
tween the  two  points  (p^,  O^)  and  (p2,  6^' 

Since  -^  =  ae«*, 

it  follows  that  p  —  =  -, 

dp     a 


and  length  of  arc  =  y'\\  -\-ldp  =  Va-^  +  1  (pj  -  p,). 

Ex.  2.   Find  the  length  of  arc  of  the  cardioid  p  =  a  (1  -  cos  ^). 

Ex.  3.   Find  the  length  of  the  cissoid  p  =  2  a  tan  6  sin  0  from  0  =  0 
to  6  =  '^- 

Ex.  4.   Find  the  entire  length  of  the  curve  p  =  2a  sin 9.    ^ 

Ex.  5.   Find  the   length  of  the  parabola  pz=i  a  sec^  ^     between  the 
points  (p,,  d,)  and  (pg,  6.^). 

Ex.  6.   Find  tlie  length  of  arc  of  the  hyperbolic  spiral  pd  =  a  between 
the  points  (p,,  ${)  and  (p.^,  $^). 


V^ 


158-159.]  GEOMETRICAL  APPLICATIONS  273 

159.  Measurement  of  arcs  by  the  aid  of  parametric  repre- 
sentation. When  the  coordinates  of  a  point  on  a  given 
curve  can  be  conveniently  expressed  in  terms  of  a  variable 
parameter,  the  problem  of  calculating  its  length  is  often 
simplified. 

Suppose  a  curve  has  its  rectangular  coordinates  expressible 
in  the  form 

in  which  <^(t)^  "^(f)  ^^^  single-valued  functions  of  the  vari- 
able U     Then 

dl 
dy  _dt      -J    _d^  ;,x 
dx      dx^         ""  dt     ' 
dt 

and  formula  (6),  p.  270,  becomes 


rv(i 


in  which  t\  t'^  are  the  values  of  t  corresponding  to  the  ex- 
tremities of  the  arc  whose  length  is  to  be  found. 

In  like  manner,  if  (p,  6)  are  expressible  in  terms  of  a  third 
variable  f,  formula  (7),  p.  272,  becomes 


j:m'<ih 


/     Ex.  1.  Find  the  length  of  a  complete  arch  of  the  cycloid 

X  =  a{0  —  sin  ^), 

y  =  a(l  —  cos  6). 
V     Ex.  2.   Find  the  length  of  the  epicycloid 

X  =  a(m  cos t  —  cos mt),  y  =  a(m  sin  t  —  sin  m{) 

from  t  =  0  to  t=  -?^!^. 

m  —  1 

^       Ex.  3.   Find  the  length  of  the  hypocycloid  x^  -\-  yf  =  as  by  expressing 
X  and  y  in  the  form        x  =  a  cos^O,  y  =  a  sin^  0. 


274 


INTEGRAL   CALCULUS 


[Ch.  VII. 


'  Ex.  4.   Find  the  length  of  the  involute  of  the  circle 

X  =  a(cos  0  -\-$sin6)f  y  =  a(sin  ^  —  ^  cos  0) 

.  from  ^  =  0  to  ^  =  ^1- 

y^    Ex.  5.    Find  the  length  of  arc  of  the  curve  x^  —  y^  =  a^,  from  (a,  0) 

to  i^v  Vi)' 

Assume  a;  =  asec^^,  ?/ =  a  tan^^. 

\^Ex.  6.   Find  the  length  of  arc  of  the  ellipse  ^  +  3^  =  1. 


Putting  X  =  acos<f>,  y  =  b sin  <j>, 

complete  arc  =  J     Vl  —  e^  cos^  ^  d<^ 


=  2  ATT  [1  -  I 


.4_  ... 


64 


], 


by  expanding  v  1  —  e^  cos^  </>  into  a  series  and  integrating  term  by  term. 
y      Ex.  7.   Find  the  length  of  arc  of  the  curve 

x  —  e^  sin  6,  y  =  e^  cos  6 
from  ^  =  0  to  ^  =  ^1. 


fi^ 


160.  Area  of  surface  of  revolution.  Let  ^^  be  a  con- 
tinuous arc  of  a  curve  whose  equa- 
tion is  expressed  in  rectangular 
coordinates  x  and  y.  It  is  required 
to  determine  a  formula  for  the 
area  of  the  surface  generated  by 
revolving  the  arc  AQ  about  the 
~    a;-axis. 

It  has  been  shown  in  Art.  86, 
p.  140,  that  if  8  denotes  the  area 
of  the  surface  generated  by  the  rotation  of  AP  (P  being 
a  variable  point  with  coordinates  (rr,  y)),  then 


Fig.  74. 


dS 
ds 


=  2  7r^. 


from  which 

f— W..(|T. 

and 

f-'W.*(|)- 

159-1*60.]  GEOMETBICAL  APPLICATIONS  275 

Hence,  by  integrating  these  two  expressions, 


surface  =  2  rr,j^y^^lj^(^Jdx 

the  limiting  values  of  x  and  y  being  the  coordinates  of  the 
points  A  and  Q. 

That  the  result  of  integration  is  to  be  evaluated  between 
the  limits  a  and  h  (or  a  and  y8)  is  readily  seen  by  following 
the  suggestions  made  in  Art.  154.  For,  denoting  the 
indefinite  integral 

by  (f>  (x)  +  (7,  since  the  area  is  evidently  zero  when  a:  =  a 
(^.e.,  when  the  point  P  coincides  with  A)  it  follows  that 

</,(«) +(7=0, 
whence  C  =  —  ^(a). 

Moreover,  when  P  coincides  with  Q  the  required  surface  is 
determined,  and  therefore 

surface  generated  hy  AQ  =  <^ (J))  -\-  C=  (f)(b}—  ^(<«). 

But  according  to  Art.  148,  <^(5)  —  <^(«)  is  the  definite  inte- 
gral obtained  by  evaluating  (10)  between  the  limits  a  and  b. 
In  like  manner  it  is  found  that  the  area  of  the  surface 
obtained  by  revolving  AQ  about  the  2/-axis  is 


In  each  of  the  above  cases  there  is  a  choice  of  two  formu- 
las for  the  area.  That  one  should  be  selected  which  can 
most  easily  be  integrated. 


276  INTEGRAL   CALCULUS  [Ch.  VII. 

Ex.   1.  Find  the  surface  of  the  catenoid  obtained  by  revolving  the 

X  X 

catenary  y  =  ^  (e*  +  e  **)  about  the  y-axis,  from  x  =  Otox  =  a, 

Since  ^  =  Ke*-e~«), 

dx 

it  follows  that 

\dxl  4 

and  hence,  by  using  the  second  formula  "of  (11),  the  required  surface  has 
the  area 


2  IT  J  a;(e«  +  e   <^)dx. 

Jo 


Ex.  2.  Find  the  surface  obtained  by  revolving  about  the  y-axis  the 
quarter  of  the  circle  x^  +  y^+ 2 x  +  2y  -h  1  =  0  contained  between  the 
points  where  it  touches  the  coordinate  axes. 

Ex.  3.  Find  the  surface  generated  by  revolving  the  parabola  y^  =  ipx 
about  the  a:-axis  from  the  origin  to  the  point  (p,  2p). 

Ex.  4.  Find  the  surface  generated  by  the  revolution  about  the  y-axis 
of  the  same  arc  as  in  Ex.  3. 

Ex.  5.    Find  the  surface  generated  by  the  revolution  of  the  ellipse 

(a)  about  its  major  axis  (the  prolate  spheroid) ; 
(6)  about  its  minor  axis  (the  oblate  spheroid). 

Ex.  6.  Find  the  surface  generated  by  the  revolution  of  the  cardioid 
p  =  a(l  4-  cos  6)  about  the  polar  axis. 

Regarding  the  figure  as  referred  in  the  first  place  to  rectangular  axes 
such  that  x  —  p  cos  B,  y  =  p  sin  0  we  have 

surface  =  2  tt  f  y  ds  =  2  tt  f  "psin^-ypHf-^Vrf^, 
since  ds  =^l^  +  (^^dO  by  Art.  87. 

Ex.  7.   Find  the  surface  of  the  cone  obtained  by  revolving  that  portion 

of  the  line  -  +  ^  =  1  which  is  intercepted  by  the  coordinate  axes, 
a      b 

(a)  about  the  X-axis; 

(^)  about  the  ^-axis. 


160-161.] 


GEOMETRICAL  APPLICATIONS 

/ 


277 


Ex.  8.   Find  the  surface  of  the  sphere  obtained  by  revolving  the  circle 
p  =  2a  cos  0  about  the  polar  axis.     [Cf .  Ex.  6.] 

Ex.  9.   Find  the  surface  generated  by  the  revolution  of  a  complete  arch 
of  the  cycloid  x  =  a{$—  sinO),  ^/  =  a(l  —  cos^)  about  the  a:-axis. 


Fig.  75. 


161.  Volume  of  solid  of  revolution.  Let  the  plane  area, 
bounded  by  an  arc  FQ  of  a  given  curve  (referred  to  rectangular 
axes)  and  the  ordinates  at 
the  extremities  P  and  Q,  be 
revolved  about  the  a;-axis. 
It  is  required  to  find  the 
volume  of  the  solid  so 
generated. 

Let  the  figure  AFQB 
be  divided  into  n  strips  of 
width  Ax  by  means  of  the 
ordinates  A^P^,  -^2^2'  ***' 
^n-iPn-i'  In  revolving 
about  the  a;-axis,  the  rec- 
tangle APR^A^  generates  a  cylinder  of  altitude  Ax,  the  area 
of  whose  base  is  ir  .  AJ^.     Hence 

volume  of  cylinder  =  tt  •  AP  •  Ax. 

The  volume  of  this  cylinder  is  less  than  that  generated  by 
the  strip  APP^A^  by  the  amount  contained  in  the  ring  gen- 
erated by  the  triangular  piece  PR^P^  Imagine  this  ring 
pushed  in  the  direction  parallel  to  the  ic-axis  until  it  occupies 
the  position  of  the  ring  generated  by  QBE.  If  every  other 
neglected  portion  (such  as  is  generated  by  Pi_iPiRi)  is 
treated  in  like  manner,  it  is  evident  that  their  sum  is  less 
than  the  volume  generated  by  the  strip  A„_iP„_iQB,  and 
hence  has  zero  for  limit  as  Ax  approaches  zero.     Therefore 


278  INTEGRAL   CALCULUS  [Ch.  VII. 

the  sum  of  the  n  cylinders  generated  by  the  interior  rectan- 
gles of  the  plane,  viz., 

ttCAP  +  A^^  +  ...  +  A^^P,J)^x, 

has  for  limit  the  volume  required.     But  the  limit  of  this  sum 

is  [by  formula  (5),  p.  250]  the  definite  integral  j   iry^dx^  and 

hence  ^6 

volume  =  TT  I   y^dx. 

The  same  result  is  readily  obtained  by  integrating  the 
expression  for  the  derivative  of  volume.      [Art.  85^  p.  140.] 

The  volume  generated  by  revolution  about  the  «^-axis  is 
found  by  a  like  process  to  be  expressed  by  the  definite 
integral 

in  which  a  and  fi  are  the  values  of  y  at  the  extremities  of 
the  given  arc. 

Ex.  1.  Find  the  volume  of  the  oblate  spheroid  obtained  by  revolving 
the  ellipse  -^  +  ^  =  1  about  its  minor  axis. 

Ex.  2.  Find  the  volume  of  the  sphere  obtained  by  revolving  the 
circle  x^  ■\-  {y  —  ky  =  r^  about  the  y-axis. 

Ex.  3.  The  arc  of  the  hyperbola  xy  =  k^,  extending  from  the  vertex 
to  infinity  is  revolved  about  its  asymptote.     Find  the  volume  generated. 

What  is  the  volume  generated  by  revolving  the  same  arc  about  the 
other  asymptote  ? 

Ex.  4.  Find  the  entire  volume  obtained  by  rotating  the  hypocycloid 
X*  +  y>  =  as  about  either  axis. 

Ex.  5.  Find  the  volume  obtained  by  the  revolution  of  that  part  of 
the  parabola  Vx  -f-  y/y  =  y/a  intercepted  by  the  coordinate  axes  about 
one  of  those  axes. 
J     Ex.  6.   Find  the  volume  generated  by  the  revolution  of  the  witch 

y  = 2^ — .  about  the  x-axis. 

^     a«+4a2 


161-162.]  GEOMETRICAL  APPLICATIONS  279 

Ex.  7.  Find  the  volume  generated  by  the  revolution  of  the  witch 
about  the  ?/-axis,  taking  the  portion  of  the  curve  from  the  vertex  (x  =  0) 
to  the  point  (x^,  y^). 

What  is  the  limit  of  this  volume  as  the  point  (x^,  y^  moves  tov^ard 
infinity  ? 

Ex.  8.  Find  the  volume  obtained  by  revolving  a  complete  arch  of  the 
cycloid  a;  =  a (^  —  sin  ^),  y  =  a(l  —  cos  &)  about  the  ar-axis. 

Volume  =  TT  f  ''''y^dx  =  Tra^  f  ''(I  -  cos  dydO. 

Ex.  9.  Find  the  volume  obtained  by  revolving  the  cardioid 
p  =  a{l  —  cos  6)  about  the  polar  axis. 

Assume  a;  =  p  cos  ^,    ^  =  p  sin  6. 

Then         dx=:  d(p  cos  6)  =  d[a{l  —  cos  0)  cos  $]  " 

=  asin^(-l  +  2cos^)<f^. 
Hence 

Volume  =  7r( y^dx  =  -  7ra^Csin»0(l  -  cos  $y(l  -  2  cos  e)d$. 

Ex.  10.  A  quadrant  of  a  circle  revolves  about  its  chord.  Find  the 
volume  of  the  spindle  so  generated. 

The  equation  of  the  circle  being  taken  in  the  form 

the  a:-axis  can  be  assumed  as  the  axis  of  rotation.    The  ordinates  of  the 
rotated  arc  are  determined  by  the  formula 


^  162.  Miscellaneous  applications.  In  the  preceding  article 
the  volume  of  the  solid  of  revolution  is  shown  to  be  the 
limit  of  the  sum  of  the  volumes  of  a  series  of  cylindrical 
plates  of  thickness  Lx.  The  notion  here  involved  is,  with 
suitable  modifications,  applicable  to  a  variety  of  problems. 
The  following  examples  (excepting  Exs.  6,  9,  10)  are  illus- 
trations of  this  principle. 


280  INTEGRAL   CALCULUS  [Ch.  VII. 

Ex.  1.  Find  the  volume  of  the  ellipsoid  ^  +  ^  +  5?  =  1. 

^         a^     h^     c^ 

Imagine  the  solid  divided  into  a  number  of  thin  plates  by  means  of 
planes  perpendicular  to  the  x-axis  and  at  equal  distances  Aa;  from  each 
other.  Regard  the  volume  of  each  plate  as  approximately  that  of  an 
elliptic  cylinder  of  altitude  Aa;.  The  base  of  the  cylinder  will  be  the 
ellipse  in  which  the  ellipsoid  is  intersected  by  one  of  the  cutting  planes. 
If  the  equation  of  this  plane  be  denoted  by  a:  =  A,  the  equation  of  the 
elliptic  base  of  the  cylinder  is  (in  y,  z  coordinates) 

or  j^—^ +  -7—^ =1. 

The  semi-axes  of  the  ellipse  are 

-v^^^rx"2     and     ^v/^TTlp. 
a  a 

Since  the  area  of  the  ellipse  is  the  product  of  the  semi-axes  multiplied 
by  TT  (Ex.  13,  p.  263),  it  follows  that 

area  of  elliptic  base  =  tt  .  -  Va^  -  A^  •  -  Va^  -  A* 
a  a 

=  ^(a'»-A2), 

and  volume  of  elliptic  cylinder  =  ^^  (a^  —  A')  AA, 

(AA  being  used  in  place  of  Aar  since  x  =  A). 

The  result  of  summing  all  such  terms  and  taking  the  limit  as  AA  ap- 
proaches zero  is  equivalent  to  the  definite  integral 

£^  l±^(a^  _  Xi)dX  =  2^'  ^  (a«  -  A'')  dX, 

On  account  of  the  ellipsoid  being  symmetrical  with  respect  to  the 
plane  x  =  0,  the  limits  0  and  a  include  one  half  the  required  volume 
and  hence  instead  of  using  limits  —a  and  -f- a  it  is  more  convenient  to 
write  the  definite  integral  in  the  above  form. 

Ex.  2.   Find  by  the  method  of  Ex.  1  the  volume  of  the  elliptic  cone 


1)'. 


measured  from  the  yz-plane  as  base  to  the  vertex  (1,  0,  0). 

Ex.  3.   Find  the  volume  of  a  pyramid  of  altitude  h  and  of  base-area  A, 


162.] 


GEOMETRICAL  APPLICATIONS 


281 


Ex.  4.   Given  an  ellipse    ^  +  ^  =  1.     On  the  major  axis  a  plane 

rectangle  A  BCD  is  con- 
structed perpendicular  to 
the  plane  of  the  ellipse. 
Through  any  point  P  of 
the  line  CD  a  plane  is  con- 
structed perpendicular  to 
CD.  The  two  points  R 
and  S  in  which,  the  latter 
plane  meets  the  ellipse  are 
joined  to  P  by  straight 
lines.  The  totality  of  all 
lines  so  determined  form 
a    ruled    surface    called    a  ^^^'  '^^' 

conoid.     Given  AC  =pj  find  the  volume  of  the  above  conoid. 

Ex.  5.  A  rectangle  moves  from  a  fixed  point  P  parallel  to  itself,  one 
side  varying  as  the  distance  from  P,  and  the  other  as  the  square  of  this 
distance.  At  the  distance  of  2  ft.,  the  rectangle  becomes  a  square  of  3 
ft.  on  each  side.     What  is  the  volume  generated  ? 

Ex.  6.  A  string  ^J3  of  length  a  has  a  weight  attached  at  B.  The 
other  extremity  A  moves  along  a  straight  Une  OX  drawing  the  weight 

r 


A 
Fig.  77. 


in  a  rough  horizontal  plane  XOY.     The  path  traced  by  the  point  B  is 
called  the  tractrix.    What  is  its  equation  ? 


282  INTEGRAL   CALCULUS  [Ch.  VII. 

Let  OF  be  the  initial  position  of  the  string  and  AB  any  intermediate 
position.  Since  at  every  instant  the  force  is  exerted  on  the  weight  B  in 
the  direction  of  the  string  BA,  the  motion  of  the  point  must  be  in  the 
same  direction ;  that  is,  the  direction  of  the  tractrix  at  B  is  the  same  as 
that  of  the  line  BA  and  hence  BA  is  a  tangent  to  the  curve.  The 
expression  for  the  tangent  length  is  (Art.  79,  p.  130) 


dx 


Solving  for    -—, 
dy 


dx 


=  V"- 


dy       ^      2/^ 
Integrating  with  respect  to  y, 


=1 


^  dy  =  V  a^  -  y'^  —  a  log —  +  C. 


The  constant  of  integration  is  determined  by  the  assumption  that  (0,  a) 
is  the  starting  point  of  the  curve.  Substituting  these  coordinates  in  the 
above  equation  we  find  C  =  0. 

Ex.  7.  A  woodman  fells  a  tree  2  feet  in  diameter,  cutting  halfway 
through  on  each  side.  The  lower  face  of  each  cut  is  horizontal,  and  the 
upper  face  makes  an  angle  of  60°  with  the  lower.  How  much  wood  does 
he  cut  out? 

Ex.  8.  The  center  of  a  square  moves  along  a  diameter  of  a  given  circle 
of  radius  a,  the  plane  of  the  square  being  perpendicular  to  that  of  the 
circle,  and  its  magnitude  varying  in  such  a  way  that  two  opposite  vertices 
move  on  the  circumference  of  the  circle.  Find  the  volume  of  the  solid 
generated. 

Ex.  9.  The  equiangular  spiral  is  a  curve  so  constructed  that  the  angle 
between  the  radius  vector  to  any  point  and  the  tangent  at  the  same 
point  is  constant.     Find  its  equation. 

Ex.  10.  Determine  the  curve  having  the  property  that  the  line  drawn 
from  the  foot  of  any  ordinate  of  the  curve  perpendicular  to  the  cor- 
responding tangent  is  of  constant  length  a. 


162.] 


GEOMETRICAL  APPLICATIONS 


283 


If  the  angle  which  the  tangent  makes  with  the  a:-axis  be  denoted  by 
<^,  it  is  at  once  evident  that 


y 


=  cos  <f> 


V1  +  taii2</> 


V 


1+f^y 


\dx, 


From  this  follows 


log(y+-\/y^-a^)  +  a 


When  the  tangent  is  parallel 
to  the  a:-axis  the  ordinate  itself 
is  the  perpendicular  a.  If  this 
ordinate  be  chosen  for  the  y-axis 
the  point  (0,  a)  is  a  point  of  the 
curve,  and  hence 

C  =  —  log  a. 

The  equation  can  accordingly 
be  written 


Fig.  78. 


(1) 


y  +  y/f 


From  this  follows,  by  taking  the  reciprocal  of  both  members, 


=  e  «> 


or,  rationalizing  the  denominator, 

(2)  ^ ^ =  e  «. 

^  ^  a 

2 
Adding  (1)  and  (2)  and  dividing  by  -> 

*  _x 

y  =  ^(e'a  +  e    «), 

which  is  the  equation  of  the  catenary. 

Ex.  11.  A  right  circular  cone  having  the  angle  2  ^  at  the  vertex  has 
its  vertex  on  the  surface  of  a  sphere  of  radius  a  and  its  axis  passing 
through  the  center  of  the  sphere.  Find  the  volume  of  the  portion  of  the 
sphere  which  is  exterior  to  the  cone. 

X^        ^2 

Ex.  12.  Find  the  volume  of  the  paraboloid  —-\-^=z  cut  off  by  the 
plane  z  =  c.  ^ 


284  INTEGRAL   CALCULUS  £Ch.  VII. 

EXERCISES  ON  CHAPTER  VII 

1.   Find  the  area  bounded  by  the  hyperbola  xy  —  a\  the  a:-axis,  and 
the  two  ordinates  x  =  a,  x  =  na. 

From  the  result  obtained,  prove  that  the  area  contained  between  an 
infinite  branch  of  the  curve  and  its  asymptote  is  infinite. 

Cj    2.   Find  the  area  contained  between  the  curves  y^  =  x  and  a^  =  y, 

3.  Find  the  area  of  the  evolute  of  the  ellipse 

(ax)^  +  (by)^  =  (a"^  -  b^)l 

4.  Find  the  area  bounded  by  the  parabola  Vx  +  y/y  =  Va  and  the 
coordinate  axes. 

5.  Find  the  area  contained  between  the  curve 

^        a  +  x 
and  its  asymptote  x  =  —  a. 

[Hint.     The   integration  may  be   facilitated   by  the   substitution 
X  =  a  cos  $.'] 

6.  Find  the  area  between  the   curve   y\y^  —  2)  =  a:  —  1  and  the 
coordinate  axes» 

V  7.  Find  the  area  common  to  the  two  ellipses 

8.  If  (a,  a)  and  (6,  P)  are  two  pairs  of  values  of  x  and  y,  the 
formula  for  integration  by  parts  gives 

J   y  dx  =  bp  —  aa  ~  i   xdy. 

Interpret  this  result  geometrically  in  terms  of  area. 

9.  Find  the  area  bounded  by  the  logarithmic  (or  equiangular)  spiral 
p  =  e««  and  the  two  radii  p^,  p^ 

10.   Find  the  length  of  an  arc  of  the  spiral  of  Archimedes  p  —  aO 
f     between  the  points  (pj,  ^,),  (pg,  6^. 

J      11.   Find  the  surface  of  the  ring  generated  by  revolving  the  circle 
a:2  +  (y  -  ky  =  a^  {k>d)  about  the  a>axi8. 

12.  Find  the  volume  of  the  ring  defined  in  Ex.  11. 


162.]  GEOMETRICAL  APPLICATIONS  285 

13.  Find  the  volume  obtained  by  revolving  about  the  a;-axis  that 

X  X 

portion  of  the  catenary  ^  ~^ (e"  +  e~*) 

limited  by  the  points  (—x^,  y^)  and  (xj,  y^, 

14.  Find  the  entire  volume  generated  by  the  revolution  of  the  cissoid 


a  —  X  X  / 

about  its  asymptote.  i    \  * 

[Hint.     For  the  purpose  of  integration,  assume  tyt* 

x  =  2a  sin2  6,  whence  y  =  1^1^. 1 
cos  6     J 

15.  Find  the  surface  generated  by  the  rotation  of  the  involute  of  the 
^  circle  x  =  a(cos  t  +  t  sin  t),  y  =  a(sin  t  —  t  cos  t) 

about  the  a:-axis  from  t  =  0\>o  t  =  ty 

16.  Find  the  volume  generated  by  the  revolution  of  the  tractrix  (see 
Ex.  6,  p.  281)  about  the  positive  ar-axis. 

17.  Find  the  area  of  the  surface  of  revolution  described  in  Ex.  16. 

,  /    18.  Find  the  length  of  the  tractrix  from  the  cusp  (the  point  (0,  o)) 
to  the  point  (xj,  y^). 


CHAPTER   VIII 

SUCCESSIVE  INTEGRATION 

163.  Functions  of  a  single  variable.     Thus  far  we  have 
considered  the  problem  of  finding  the  function  y  oi  x  when 

-^  only  is  given.  It  is  now  proposed  to  find  y  when  its  nth 
derivative  -^  is  given. 

The  mode  of  procedure  is  evident.     First  find  the  func- 

tion       ^_^  which  has  -y^  for  its  derivative.     Then,  by  inte- 

d^'-^y 
grating  the  result,  determine    ■,  n_2,  and  so  on  until  after  n 

successive  integrations  the  required  result  is  found.  As  an 
arbitrary  constant  should  be  added  after  each  integration  in 
order  to  obtain  the  most  general  solution,  the  function  y  will 
contain  n  arbitrary  constants. 

Ex.  1.     Given  ^4  =  -«»  ^^^  V' 

dx^     x^  ^ 

Integration  of  —  with  respect  to  x  gives 
3r 


H^o+C'^ 


dx^~     2x2 
Integrating  a  second  time, 

dx     2x         *  * 

and  finally  y  =  i  log  a:  +  i  C^x^  +  CgX  +  C^ 

The  triple  integration  required  in  this  example  will  be  symbolized  by 

which  will  be  called  the  triple  integral  of  —  with  respect  to  x. 


Ch.  VIII.  163.]         SUCCESSIVE  INTEGRATION  287 

Ex.  2.  Determine  the  curves  having  the  property  that  the  radius  of 
curvature  at  any  point  P  is  proportional  to  the  cube  of  the  secant  of  the 
angle  which  the  tangent  at  P  makes  with  a  fixed  line. 

If  a  system  of  rectangular  axes  be  chosen  with  the  given  line  for 
a;-axis,  it  follows  from  equation  (6),  p.  164,  and  from  Art.  10,  that 


in  which  a  is  an  arbitrary  constant.     This  equation  reduces  to 

d^  y 


dx^ 
from  which  follows 


:^a, 


y  =  I  Ja(rfx)2  =?  a[|'  +  C^x  +  C2], 


C,  and  C2  being  constants  of  integration.     Hence  the  required  curves 
are  the  parabolas  having  axes  parallel  to  the  y-axis. 

The  existence  of  the  two  arbitrary  constants  Cj,  Cg  in  the  preceding 
equation  makes  it  possible  to  impose  further  conditions.  Suppose,  for 
example,  it  be  required  to  determine  the  curve  having  the  property 
already  specified,  and  having  besides  a  maximum  (or  a  minimum)  point 
at  (1,  0). 

Since  at  such  a  point  -^  =  0,  it  follows  that 
dx 

0  =  a(l  +  Ci), 
whence  Cj  =  —  1. 

Also,  by  substituting  (1,  0)  in  the  equation  of  the  curve, 

0  =  a(i-l  +  C2), 
from  which  Cg  =  i- 

Accordingly  the  required  curve  is 

Ex.  3.  Find  the  equation  (in  rectangular  coordinates)  of  the  curves 
having  the  property  that  the  radius  of  curvature  is  equal  to  the  cube  of 
the  tangent  length. 

[Hint.     Take  y  as  the  independent  variable.] 


288  INTEGRAL   CALCULUS  [Ch.  Vlll. 

Ex.  4.  A  particle  moves  along  a  path  in  a  plane  such  that  the  slope 
of  the  line  tangent  at  the  moving  point  changes  at  a  rate  proportional  to 
the  reciprocal  of  the  abscissa  of  that  point.  Find  the  equation  of  the 
turve. 

Ex.  5.  A  particle  starting  at  rest  from  a  point  P  moves  under  the 
action  of  a  force  such  that  the  acceleration  (cf.  Ex.  14,  p.  Ill)  at  each 
instant  of  time  is  proportional  to  (is  k  times)  the  square  root  of  the  time. 
How  far  will  the  particle  move  in  the  time  t'i 

164.  Integration  of  functions  of  several  variables.     When 

functions  of  two  or  more  variables  are  under  consideration, 
the  process  of  differentiation  can  in  general  be  performed 
with  respect  to  any  one  of  the  variables,  while  the  others 
are  treated  as  constant  during  the  differentiation.  A  repe- 
tition of  this  process  gives  rise  to  the  notion  of  successive 
partial  differentiation  with  respect  to  one  or  several  of  the 
variables  involved  in  the  given  function.     [Cf.  Arts.  68,  72.] 

The  reverse  process  readily  suggests  itself,  and  presents 
the  problem  :  Griven  a  partial  (^first^  or  higher)  derivative  of  a 
function  of  several  variables  with  respect  to  one  or  more  of  these 
variables^  to  find  the  original  function. 

This  problem  is  solved  by  means  of  the  ordinary  processes 
of  integration,  but  the  added  constant  of  integration  has  a 
new  meaning.     This  can  be  made  clear  by  an  example. 

Suppose  u  is  an  unknown  function  of  x  and  y  such  that 

dx 

Integrate  this  with  respect  to  x  alone,  treating  y  at  the 
same  time  as  though  it  were  constant.     This  gives 

in  which  ^  is  an  added  constant  of  integration.  But  since 
y  is  regarded  as  constant  during  this  integration  there  is 
nothing  to  prevent  <f>  from  depending  on  it.     This  depend- 


163-165.]  SUCCESSIVE  INTEGRATION  289 

ence  may  be  indicated  by  writing  <^(y)  in  the  place  of  (j). 
Hence  the  most  general  function  having  2  a:  4-  2  ?/  for  its 
partial  derivative  with  respect  to  x  is 

U  =  X^+2X7/  -^(l>(7/}, 

in  which  <^(y)  is  an  entirely/  arbitrary  function  of  y. 
Again,  suppose 


dxdy 

Integrating  first  with  respect  to  y^  x  being  treated  as 
though  it  were  constant  during  this  integration,  we  find 

where  '>^(x)  is  an  arbitrary  function  of  x^  and  is  to  be 
regarded  as  an  added  constant  for  the  integration  with 
respect  to  y. 

Integrate  the  result  with  respect  to  a;,  treating  y  as  con- 
stant.    Then 

Here  <!>(«/),  the  constant  of  integration  with  respect  to  x^ 
is  an  arbitrary  function  of  «/,  while 

'^(x)=^'>\r(x)dx. 

Since  '>^(x)  is  an  arbitrary  function  of  x^  so  also  is  "^(x). 

165.  Integration  of  a  total  differential.  The  total  differen- 
tial of  a  function  u  depending  on  two  more  variables  has 
been  defined  (Art.  69)  by  the  formula 

du=^^dx^^-^dy. 
dx         dy  ^ 

The  question  now  presents  itself:  Given  a  differential 
expression  of  the  form 

Fdx  +  Qdy,  (1) 


290  INTEGRAL   CALCULUS  [Ch.  VIII. 

wherein  P  and  Q  are  functions  of  x  and  y^  does  there  exist 
a  function  u  of  the  same  variables  having  (1)  for  its  total 
differential  P 

It  is  easy  to  see  that  in  general  such  a  function  does  not 
exist.  For,  in  order  that  (1)  may  be  a  total  differential  of  a 
function  «*,  it  is  evidently  necessary  that  P  and  Q  have  the 
form 

P  =  ^,   Q  =  ^.  (2) 

dx  By 

What  relation,  then,  must  exist  between  P  and  Q  in  order 
that  the  conditions  (2)  may  be  satisfied  ?  This*  is  easily 
found  as  follows  :  Differentiate  the  first  equation  of  (2)  with 
respect  to  y^  and  the  second  with  respect  to  x.     This  gives 

dP^^u_    dQ ^  d^u 
dy      dydx    dx      dxdy 
from  which  follows 

dP^8Q  (g. 

dy      dx  ^  ^ 

This  Is  the  relation  sought. 

The  next  step  is  to  find  the  function  u  by  integration.  It 
is  easier  to  make  this  process  clear  by  an  illustration. 

Given      (2x  +  2y-\-2')dx+(2y  +  2x-\-2)dy, 
find  the  function  u  having  this  as  its  total  differential. 

Since         P=2a;  +  2y  +  2,   Q  =  2y  +  2x  +  2, 
it  is  found  by  differentiation  that 

^  =  2  and  ^=2, 
dy  dx 

and  hence  the  necessary  relation  (3)  is  satisfied. 
From  (2)  it  follows  that 

|^=2a;  +  2y  +  2. 
dx 


165.]  SUCCESSIVE  INTEGRATION  291 

Integrating  this  with  respect  to  x  alone, 

u  =  x^-{-2xy  +  2x  +  (l>(iy).  (4) 

It  now  remains  to  determine  the  function  <^(^)  so  that 

^(=(?)=^^  +  2^  +  2.  (5) 

Differentiate  (4)  with  respect  to  y  alone,  whence 

dy 

where  <j>\y)  denotes  the  derivative  of  <^C^)  with  respect  to 
y.     The  comparison  of  this  result  with  (5)  gives 

2y4-2r?:+2  =  2rr  +  </)'C«^), 

or  <^'(^)=2^  +  2,  (6) 

whence,  by  integrating  with  respect  to  y^ 

Ky^^y^+'iy  +  O, 

in  which  O  is  an  arbitrary  constant  with  respect  to  both 
X  and  y. 

Hence  u  =  x^-\-2xy-\-2x-iry'^  +  2y  +  Q. 

It  is  to  be  remarked  that  in  integrating  (6)  we  integrate 
exactly  those  terms  in  Q  which  do  not  contain  x.  Hence 
the  following  rule  may  be  formulated  for  integrating  a  total 
differential : 

Integrate  P  with  respect  to  x  alone^  treating  y  as  constant. 
Then  integrate  with  respect  to  y  those  terms  of  Q  which  con- 
tain y  hut  do  not  contain  x^  and  add  the  result^  together  with 
an  arbitrary  constant  0,  to  the  terms  already  obtained. 

It  is  evident  that  it  would  be  equally  well  to  first  inte- 
grate Q  with  respect  to  ?/,  and  then  integrate  those  terms 
of  P  which  contain  x  alone  with  respect  to  x,  and  add  the 
two  results. 


^. 


292  INTEGRAL   CALCULUS  [Ch.  VIII. 


EXERCISES 

Determine  in  each  of  the  following  cases  the  function  u  having  the 
given  expression  for  its  total  differential : 

1.  ydx  +  x  dy. 

2.  sin  X  cos  ydx+  cos  a;  sin  y  dy. 

3.  ydx  —  X  dy. 

ydx  —  xdy  ' 

xy 
5.   (3a;2  -  3 ay) dx  +  (3y2  _  ^ax)dy. 

ydx  xdy 

'  ar2+3/2     y2+a.2* 

7.  {2x^-\-2xy-\-b)dx  +  {x^  +  y^-y)dy. 

8.  (a:*  +  !/*  +  a;2  -  y^)dx  -\-  (^y^x  -  2xy  +  y  -  y^+  2)dy. 


166.   Multiple   integrals.      The   integration   of  was 

considered  in  Art.  164.  If  F(^x,  y)  be  written  for  the  given 
function,  the  required  integration  will  be  represented  b} 
the  symbol 

u  =  j  JFQx,  i/)dxdi/, 

and  the  function  sought  will  be  called  the  double  integral  of 
F(x^  y)  with  respect  to  x  and  y. 


Likewise  III  -^(^'  ^'  z)dx  dy  dz 


will  be  called  the  triple  integral  of  F(x^  y»  2)'     It  represents 

the  function  t*  whose  third  partial  derivative  — — — --  is  the 

ox  ay  dz 

given  function  F(x,  y,  «).  It  will  be  understood  in  what 
follows  that  the  order  of  integration  is  from  right  to  left, 
that  is,  we  integrate  first  with  respect  to  the  right  hand  vari- 
able 2,  then  with  respect  to  y,  and  lastly  with  respect  to  x. 

Such  integrals  (double,  triple,  etc.)  will  be  referred  to  in 
general  as  multiple  integrals. 


165-167.]  SUCCESSIVE  INTEGRATION  293 

167.  Definite  multiple  integrals.  The  idea  of  a  multiple 
integral  may  be  further  extended  so  as  to  include  the  notion 
of  a  definite  multiple  integral  in  which  limits  of  integration 
may  be  assigned  to  each  variable. 

Thus  the  integral  I  j  a^y  dx  dy  will  mean  that  ^y^  is  to 
be  integrated  first  with  respect  to  y  between  the  limits  0 
and  2.     This  gives 

^y"^  dy  —  ^i7?. 


X 


The  result  so  obtained  is  to  be  integrated  with  respect  to  x 
between  the  limits  a  and  6,  which  leads  to 


^^^^dx  =  ^(b^-a^) 


as  the  value  of  the  given  definite  double  integral. 
In  general  the  expression 


££  Fix,y-)dxdy 


will  be  used  as  the  symbol  of  a  definite  double  integral. 
It  will  be  understood  that  the  integral  signs  with  their 
attached  limits  are  always  to  be  read  from  right  to  left,  so 
that  in  the  above  integral  the  limits  for  y  are  h  and  h',  while 
those  for  x  are  a  and  a'. 

Since  x  is  treated  as  constant  in  the  integration  with 
respect  to  y,  the  limits  for  y  may  be  functions  of  x.  Con- 
sider, for  example,  the  integral  I  j  xydxdy.  The  first 
integration  (with  respect  to  y)  gives 

a?  — a? 


£.,i3/  =  -[£,=  <f-|)  = 


By  integrating  this  result  with  respect  to  x  between  limits 
0  and  1  the  given  integral  is  found  to  have  the  value  —  ^^. 


/ 


294 


INTEGRAL   CALCULUS 


[Ch.  VIII. 


EXERCISES 
Evaluate  the  following  integrals : 


1.  yj^''sec\xy)dxdy. 

rn  ra{l+coa0) 

Jo  Jo 


r^  8m  edO  dr. 


rb  /•lOy      

•  Jojy   ^^y-y^dydx. 

n'  r^-^xdzdxdy 
>  Jo         x^  +  y^    ' 


168.  Plane  areas  by  double  integration.  The  area  bounded 
by  a  plane  curve  (or  by  several  curves)  can  be  readily  ex- 
pressed in  the  form  of  a  definite  double  integral.  An  illus- 
trative example  will  explain  the  method. 

Ex.  1.  Find  by  double  integration  the  area  of  the  circle  (a:  —  ay  + 
(y-6)2=r2. 

Imagine  the  given  area  divided  into  rectangles  by  a  series  of  lines 

parallel  to  the  y-axis  at 
equal  distances  Ax,  and 
a  series  of  lines  parallel 
to  the  ar-axis  at  equal 
distances  Ay. 

The  area  of  one  of 
these  rectangles  is  Ay  • 
Ax.  This  is  called  the 
element  of  area.  The 
sum  of  all  the  rectangles 
interior  to  the  circle  will 
be  less  than  the  area 
required  by  the  amount 
X  contained  in  the  small 
subdivisions  which  bor- 
der the  circumference  of 
the  circle.  By  a  method  exactly  analogous  to  that  used  in  Art.  149,  it 
is  easy  to  show  that  the  sum  of  these  neglected  portions  has  a  zero 
limit  when  Ax  and  Ay  are  both  made  to  approach  zero. 

To  find  the  value  of  the  limit  of  the  sum  of  all  the  rectangles  within 
the  circle  it  is  convenient  to  first  add  together  all  those  which  are  con- 
tained between  two  consecutive  parallels.  Let  P1P2  be  one  of  these 
parallels  having  the  direction  of  the  x-axis.     Then  y  remains  constant 


Fig.  79. 


167-169.]  SUCCESSIVE  INTEGRATION  295 


while  X  varies  from  a  —  Vr^  —  {y  —  b)'^  (the  value  of  the  abscissa  at  P,) 
to  a  +  Vr-  —  (y  —  b)'^  (the  value  at  Pg)-  The  limit  as  Ax  approaches 
zero  of  the  sum  of  rectangles  in  the  strip  from  PJ^^  i^  evidently 


(1)  Az/[limit  of  sum  (Aa;  +  Aa;  +  •.•)]  =  A?/  C-+^^^^-^y-^  ^^^ 

Now  find  the  limit  of  the  sum  of  all  such  strips  contained  within  the 
circle.  This  requires  the  determination  of  the  limit  of  the  sum  of  terms 
such  as  (1)  for  the  different  values  of  y  corresponding  to  the  different 
strips.  Since  y  begins  at  the  lowest  point  A  with  the  value  6  —  r,  and 
increases  to  6  +  r,  the  value  reached  at  B,  the  final  expression  for  the 
area  is  

\       dy  )      , dx:=^  \        \       dy  dx. 

»'h-r  •^a-Vr2-(y-6)2  ^h-r  -^ a-Vr2-{y—b)2 

Integrating  first  with  respect  to  x, 

•^a-v'r2-(y-6)2  Jo-s/r2-(y-6)2 

This  result  is  then  integrated  with  respect  to  y,  giving 
C'^''2Vr^-(y-bydy  =  (y  -b)Vr^  -  (y  -  by  +  r^  sin-i^^l  ''*"'=  irr^. 

Jh-r  f-     Jb-r 

If  the  summation  had  begun  by  adding  the  rectangles  in  a  strip  paral- 
lel to  the  2/-axis,  and  then  adding  all  of  these  strips,  the  expression  for 
the  area  would  take  the  form 


V 


X 


a+r    rb+Vr2~(x-a)2 

\      dxdy. 

r   •^6-V'r2-(x-a)2 


It  is  seen  from  this  last  result  that  the  order  of  integration  in  a  double 
integral  can  be  changed  if  the  limits  of  integration  be  properly  modified 
at  the  same  time. 

Ex.  2.   Find  the  area  which  is  included  between  the  two  parabolas 

2^2  =  9  a:  and  2/2  =  72  -  9  x.  . 

Ex.  3.   Find  the  area  common  to  the  two  circles 

a;2  _  8  a:  +  2/2  -  8  2/  +  28  =  0, 

a;2  -  8  X  +  2/2  -  4  y  +  16  =  0. 

169.  Volumes.  The  volume  bounded  by  one  or  more 
surfaces  can  be  expressed  as  a  triple  integral  when  the 
equations  of  the  bounding  surfaces  are  given. 


296 


INTEGRAL   CALCULUS 


[Ch.  vni. 


Let  it  be  required  to  find  the  volume  bounded  by  the 
surface  ABC  (Fig.  80)  whose  equation  is  z=f(x^y^^  and 
by  the  three  coordinate  planes. 

Imagine  the  figure  divided  into  small  equal  rectangular 
parallelopipeds  by  means  of  three  series  of  planes,  the  first 
series  parallel  to  the  ^2-plane  at  equal  distances  A  a;,  the 


FiQ.  80. 


second  parallel  to  the  rca-plane  at  equal  distances  Ay,  and 
the  third  parallel  to  the  iry-plane  at  equal  distances  A2. 
The  volume  of  such  a  rectangular  solid  is  AxAi/Az;  it  is 
called  the  element  of  volume.  The  limit  of  the  sum  of  all 
such  elements  contained  in  OABO  is  the  volume  required, 
provided   that   the   bounding  surface  ABQ  is  continuous. 


169.]  SUCCESSIVE  INTEGBATION  297 

(The  reader  can  easily  show  that  the  sum  of  the  neglected 
portions  is  less  than  the  volume  of  the  largest  plate  formed 
by  two  consecutive  parallel  planes  and  that  its  limit  is 
therefore  zero.) 

To  effect  this  summation,  add  first  all  the  elements  in 
a  vertical  column.  This  corresponds  to  integrating  with 
respect  to  z  (x  and  y  remaining  constant)  from  zero  to 
f(x^  ^).  Then  add  all  such  vertical  columns  contained 
between  two  consecutive  planes  parallel  to  the  ?/2!-plane  (x 
remaining  constant),  which  corresponds  to  an  integration 
with  respect  to  y  from  y  =  0  to  the  value  attained  on  the 
boundary  of  the  curve  AB.  This  value  of  y  is  found  by 
solving  the  equation  f(x^  ^)  =  0.  Finally,  add  all  such 
plates  for  values  of  x  varying  from  zero  to  the  value  at  A, 
The  final  result  is  expressed  by  the  integral 

ax  ay  az. 


in  which  ^{x)  is  the  "result  of  solving  the  equation /(a;,  ^)  =  0 
for  y^  and  a  is  the  a;-coordinate  of  A. 

Ex.  1.  Find  the  volume  of  the  sphere  of  radius  a. 
The  equation  of  the  sphere  is 

a;2  +  y^  +  2:2  _  ^2^ 


or  2  =  Va2  _  a;2  -  y\ 

Since  the  codrdinate  planes  divide  the  volume  into  eight  equal  por- 
tions, it  is  sufficient  to  find  the  volume  in  the  first  octant  and  multiply 
the  result  by  8. 

The  volume  being  divided  into  equal  rectangular  solids  as  described 
above,  the  integration  with  respect  to  z  is  equivalent  to  finding  the  limit 
of  the  sum  of  all  the  elements  contained  in  any  vertical  column.  The 
limits  of  the  integration  with  respect  to  z  are  the  values  of  z  correspond- 
ing to  the  bottom  and  the  top  of  such  a  column,  namely,  2  =  0,  and 
z  —  y/a^  —  x^  —  y\  since  the  point  at  the  top  is  a  point  on  the  surface  of 
the  sphere. 


298  INTEGRAL   CALCULUS  [Ch.  VIII.  169. 

The  limits  of  integration  with  respect  to  y  are  found  to  be  y  =  0  (the 
value  at  the  a;-axis),  and  y  =  Va^  —  x^  (the  value  of  y  at  the  circumfer- 
ence of  the  circle  a'^  —  x^  —  y^  =  0,  in  which  the  sphere  is  cut  by  the 
zy-plane). 

Finally,  the  limiting  values  for  x  are  zero  and  a,  the  latter  being  the 
distance  from  the  origin  to  the  point  in  which  the  sphere  intersects  the 
X-axis.     Hence 

F  (=  volume  of  sphere)  =  8  \    I  \  dxdydz. 

Integrating  first  with  respect  to  2, 

/•a  /'VaZ— 12      , 

F=8j^j^  yJa^-x'^-y'^dxdy\ 

then  with  respect  to  y, 

V  =  8  i''dx\y.  yJa^-x^-y^  +  "' "  ^'  sin"!        ^       l^"^^ 

47ra8 


-r 


|(a2-x2)rfx=     g 


Ex.  2.   Find  the  volume  of  one  of  the  wedges  cut  from  the  cylinder 
a:^  4-  y2  _  ^2  ijy  tiig  planes  2  =  0  and  z  =  mx. 


/ 


Ex.  3.   Find  the  volume  common  to  two  right  circular  cylinders  of 
the  same  radius  a  whose  axes  intersect  at  right  angles. 

Ex.  4.   Find  the  volume  of  the  cylinder  (a:  -  l)^  +  (y  -  1)2  =  1  limited 
by  the  plane  2  =  0,  and  the  hyperbolic  paraboloid  xy  =  2. 

Ex.  5.   Find  the  volume  of  the  ellipsoid 

a^      b^     c^ 
Ex.  6.   Find  the  volume  of  that  portion  of  the  elliptic  paraboloid 

2=1-^^-2? 

which  is  cut  off  by  the  plane  2  =  0. 


ANSWERS 


Page  20.    Art.  9 

6  a; -4.                 3.    -^. 
4a;2 

4.   4a:3_6 

37* 

Page  25.     Art.  11 

14«_4-33«2.       3.    12m2_2. 

4.   4x-5. 

1.  2  a; -2. 


1.   162/2-2. 


Page  28.     Art.  13 

2.  Inc.  from  —  co  to  ^  ;  dec.  from  |  to  1  ;  inc.  from  1  to  +  oo  ;  ^  and  1. 

3.  Two.     +l3itx=l±V^;  -I  atx  =  l±V^.  4.    ±tan-i^V 

Page  29.     Art.  14 
2.    (6u-4)6x2.  3.    -^(lOx-2).  4.    (g  n  -  A_^  (a.2  -  i). 

Page  37.     Art.  19 

10.  J4±A. 

2Vx  +  2 

J,  Va(Vx  —  Vq) ^ 

2\/x(Vx  +  a)(Va  +  \/^)2 

12.  1 

13 


1. 

10ic9. 

2. 

-8x-9. 

3. 

c 

2Vx 

4. 

1       1 

5. 

-iV^. 

6. 

w(x  +  a)'*-i. 

7. 

MX«-i. 

Vl  -  x2  (1  -  a;) 

1 


8. 

(a2 


14. 


2  xil  -  a;2)  +  VI  -  y/-^ 
ia^  +  Sx^ 


iVx^a^  +  x'^ 


2-6a;-a;2  15,  ^^       . 

(a;2  +  2)2    *  '    a;Vl-a;2 

299 


300  AN  8  WEBS 

16. 


17. 


(1  -  aj2)i(l  4-  a:2)f  '  dx 

26.  (2M  +  6a;M)^+3M2  +  4a:8. 

(x«-l)2'  wwn-i^ 

Oft  dx  nu^ 


18 

wi(6  +  a;)  + 

n{a-{-x) 

(a  +  a;)'»+i. 

(6  +  a;)«+i 

in 

-2 

x'^i:^  +  1)1 

20. 

66a;8(a;a  +  l)i 

21. 

dx 

(a  +  x)'»      (a  +  x)«+i 

27.   2  wa;8w7  ^  +  Wh^^  —  +  3  waxa^;. 
dx  dx 

jQ     —  &%  __  6x 


«'2^  «\/a^  -  x=^ 

82.    (0,0),  f-i-,~-§_V 
^    V9a       27  ay* 

22.  12(u2-t*  +  l)^.  ^^"       ^^«>' 

^  ^   ^dx  ^    (21t/8-i9zf)10x 

23.  muKl-^u^f^^  (7te2  +  5)t 

dx  35.   At  right  augles  at  (3,  ±  6). 

Page  43.    Art.  24 
1.    -4-  12.   Iogio6     2^  +  7 


as  +  a  x2  +  7a6 

__« 18. 2^ 

ax +  6  xlogx 

8x-7  14.   ae*-. 


4x2-7x  +  2  15.   4e4«+« 

2  _i_ 

r^*  16.    -=-i^. 

(H-a;)2 
4x 


1  -  X*  17. 


(1  +  e*)2 

6.  logx  +  1.  18    y_3a;2ex. 

7.  wx«-i  logx  +  x«-i.  19.    1  -  j/2. 

8.  7ix"~^  log  x«  +  mx""i.  20     e*  +  e~*  ^ 


^^--1  21.    ^^''^ 


1  x  +  e' 

10. 

2("v/x  +  1)  22.  wx"-!  a*+  ai^a*  log  a. 

11.  log.6.         12xv/2T^-l        .  23.    ^:;« 

2  V2  +  x(3  X-  -  V2  +  X)  Vx(a  -  ») 


ANSWERS 


301 


24.    - 


25. 


27. 


10. 

11. 

12. 
13. 
14. 
15. 

16. 
17. 


x([ogxy 

2  log  a; 

X 


xlogx 
x^^Qogx^  1). 

X 

Page  47. 

7  cos  7  «. 

—  5  sin  5  x. 
2  X  cos  a;2. 

2  cos  2  oj  cos  X  —  sin  2  a;  sin  x. 

3  sin2  x  cos  X. 
10  X  cos  5  x^. 

14  sin  7  a:  cos  7  a;. 
sec-^  a;  (tan2  a;  —  1). 
3  sin2  x  cos^  x  —  sin*  x. 
secx(tanx  +  secx). 

-  6  X  (1  -  2  x2)  sin  (1-2  x2)2 
cos  (1-2  x2)2. 

-20  X  (3  -  5  x2)  sec2  (3  -5  x2)2. 

2  tan  X  sec2  x  —  2  tan  x. 

secx. 

cot  Vx 

2Vx 

1  ^  ^ 

— rloga  •  ««•  sec2(a«). 

X2 

w  sin**-!  X  sin  (n  +  1)  x. 


31. 


Art. 
19. 


-(a;_l)f(7a;2  4.30x-97) 
12(x-2)i(x-3)V- 

2  +  X  -  5  x2 

2Vl  -X 
1  +  3  x2  -  2  X* 

(1  -  x2)i 
5x4(a  +  3x)2(a-2x) 
(a2  +  2ax-12x2). 

(x  —  2  g)  Vx  +  q , 
(x-a)t 

31 

?nn  sin"*-i  nx  -  cos  (m  —  n)  x 
cos'*+i  mx 
2 
1  +  tanx 


21.  csc^ 

dx 

22.  cos  (sin  u)  cos  w 


dx 


23. 
24. 

25. 

26. 

27. 
28. 

29. 
30. 

31. 


2  ae«*  sin  e"*  •  cos  e«*. 
e*  •  cos  e*  •  log  X  +  5HLi!. 


X  cos  x2 
Vsin  x2 
'sin  X 


+  cosx 


log  x^ 


—  8  csc2  4  X  cot  4  X. 

8(4  X  -  3)  sec  (4  x  -  3)2 

tan  (4  X  -  3)2. 
3  x2  sin  x^. 

sec  Vx  tan  y/x 


18.   cos  2  ?« 


dx 


-  2  X  CSC2  X2  +  ' 

y  cos  xy 
1  —  X  COS  xy 

—  CSC2  (X  +  y). 


2Vx 


4x 


Vl-X2 


Page  49.    Art. 
3. 


33 


V6  X  -  9  x2 
3 


vr 


302 


ANSWERS 


5. 
6. 

7. 
8. 

9. 

10. 

11. 

12. 

13. 

14. 
16. 
16. 


-2 

1+X2 


17.   sin-ix+- 


vT-^2 


2  Vsin-i  x  Vl  —  a^ 
1 


xVl  -(logx)2 
sec^x 


Vl  -  tan2x 
1 


2 


vT 


xV^-1 

2 
l  +  a;2* 

^  Vl  +  CSC  X. 


1ft 

gtau    z 

l  +  X^ 

19 

2 

VI -x2 

20. 

-2 
x2  +  l 

21. 

1 

2Vx(x+  1) 

22. 

-2 

e^  +  e  * 

23. 

n 

C0S2x+  W2sftl2x 

24. 

2. 

25 

2sinx 

Vl-4cos2x 

26. 

-1 

2(1  +  x2) 

27. 

-1 

sec2  X  •  tan-i  x  + 


tanx 


1  +X2 

Page  51. 


6x  +  15x2. 
-6     15 

x8        X** 

3x-l 


VT-i^ 
28.   0. 

Exercises  on  Chapter  11 


2Vir-~3 
a2  -  2  x2 
Va2  -  x2 
log  sill  X  +  X  cot  X. 


9.  — cotx. 

2Vm 

10.  1-loga. 

X 

11.  -C3x  +  x8) 

(1  +  a;2)! 

12.  c*(cos  X  -  sin  «) 
-1 


xWa^  -  y? 

2xg-2g  +  l^ 
2rx-aJ2)* 


18. 
14. 


xV2x-l 
4 


6  +  3  cos  X 
16.   tan-i-^. 


16. 


2(1  +  a^) 


AN  S  WEBS  303 


17.   4tan5a;.  26        2  xy^  +  3  x^_ 

\o<rx  '       2x'^2/  +  32/2 


19. 


(1  -  xy 

4 


2/2+1 


27. 


5  +  3COSX  -*'•    i_2a;^_a;2 


20.       ^'  28.   4  cos  (2  log  a;2- 7) 

"    1  —  «*  '  re 


29.   For  all  values. 


21. 

X  ^x 

22.    1. 


_^x_+_a^  32.   a;,    ?/    are    determined    from 

^    ^~  ^  a'^y  =z  ±b'^x  and  equation  of 


curve. 


2  tan  a;  +  e  ^^''^  •  sec  X  tan  x. 


24    ?izL«l^.  _ 

'   ax -2/2  34.   tan-i2V2» 

Pages  54-55.     Ezercises  on  Chapter  III 

^'   '^^  ^'  19.   TO"  •  cos  (  mx  +  w  -  ). 

2.  0.  \  2/ 

3.  -^-  20     (-!)-•  (m  +  n-D! 

a^  "     (w-l)!(a  +  x)'»+« 

4.  -|i.  21.   TO(- 1)^-1.  (n- 1)1. 

•^  (a  +  x)« 

5.  e^. 

6.  e^\ogx  +  ^-^' 

X  X2 

7.  2  log  X  +  3. 

8.  8  tanxsec2x(3sec2x  —  1). 

9.  2  cot  X  csc"^  X. 
10.  16  sin  x  cos  x. 

24 


3j>3 

—  2  g^xy 


XX. 

12. 

(l-x)6 
48 

X 

13. 

sin  X. 

14. 

15. 

8(ex_e-x) 

(ex  +  e-xy 

8  x2e2x. 

16. 

4! 

X2* 

17. 

a"e«^. 

18. 

r-l)«w! 

(x  -  l)«+i 


(2/2  _  ax)3 
25     -yr(^-l)^  +  (y-l)!I. 

X2(?/  -  1)3 

26.      '^~  ^    .  e^v. 
(2  -  2/)3 

32.  (- l)«.2"-i.»il 

[ 1 1 

U2x-l)'*+i      (2x+l)«+i 

33.  (^-^)^ 

X 

34    2(-l)n.ri! 
(l+.x)«+i 

36.   2»»-icos(2x+— y 


304  ANSWERS 


Page  64.     Art.  40 

6.    -8  +  4(y-3)  +  3Cy-3)2. 

Page  67.    Art.  41 

1.   x-\-—  +  —x^+—x-  +  B.  8.    1-^-^+B. 

3      15         315  2       8 

«     ,    ,  a;2     a;3     3x4_llx5  ,   „ 

"^^     I"^    8  30  9-    l+2x  +  2a;2  +  2x8+i?. 

7.    ^  +  |-  +  |'+^+^-  10.    x4  + 7x3 +11x2 -16  a; -41. 

Pages  75-76.    Exercises  on  Chapter  IV 

hi  /,3 

1.  COS  X  —  ^  sin  X cos  X  +  —  •  sin  X  +  ^. 

^  o ! 

2.  tan  /i  +  X  sec2  /^  +  x^  sec2  ;^  tan  A  +  —  sec2  A  (1  +  3  tan2  h)  +  B. 
4.    logx  +  ^-^  +  ^-^  +  i?.    ' 

^  X        2X2        3X3        4a;4^ 

6.     X6  +  5  X*y  +  10  X3y2  +  10  x22/3  4.  5  xy4  +  y6. 


Pages 

79-80.    Art.  47 

2. 

2 
a2 

a2 
+  62 

3. 
Pages 

83-84.     Art.  49 

*-^- 

8. 

f. 

8. 

-4. 

12.    12&^. 

16.   1. 

log  6 

4. 

4. 

9. 

0. 

"f- 

17.    1. 

6. 

f 

10. 

2. 

n 

18.    -f 

7. 

i. 

11. 

8. 

15.   1. 

19.    h 

Pages  87-88.    Art.  52 

1. 

1. 

6. 

log  a. 

9.   0. 

12.    -i 

t 

0. 

6. 

-f 

10.   0  or  00  according  As  n  >  0. 

8. 

0. 

7. 

4a« 

or<0. 

18.    h- 

4. 

6. 

8. 

1. 

11.   f 

14.    -1. 

ANSWERS  305 

Page  89.    Art.  54 
1.    1.  8.    e«*.  6.    1.  7.    1.  9.   e«. 


4.   1.  8_   X  «•    1- 


Pages  89-90.     Exercises  on  Chapter  V 
10.   0. 


1. 

1. 

5. 

0. 

2. 

0. 

6. 

0. 

3. 

00  if  r>l, 
Oif  r<l, 

7. 

a 

2* 

jwe'««if  r=l. 

8. 

1. 

4. 

1. 

9. 

aia2 

11. 

-1- 

12. 

a. 

13. 

1 
y/2b 

14. 

e*. 

15. 

1 

2\/2 

16. 

-  1. 

17. 

1. 

18. 

h 

an- 
Page  100.     Art.  62 

1.  _  _i.,  max. ;  J- ,  min.  «•    "  1'  °^^^-  ^  "  *'  ^i^" 

v^  v^  7.-2,  min. ;  1,  max. 

2.  2,  max.;  3,  min. 

o    o      •       a  8.    e,  max. 

3.  2,  mm.;  f,  max.  ' 

4.  (2  w  +  i)  TT,  max. ;  (2  w  +  |)  tt,         9-    2  wtt,  rain. ;  also  tan'i  ±  \/|  for 
min.  for  all  integral  values  of  n.  angles  in  2d  and  3d   quarter. 

g     a   j^.j^  (2«+l)7r,  tan-i±  V|,  Istand 

4  4th  quarter,  max. 

Pages  103-104.     Exercises  on  Chapter  VI 

1.  The  line  should  be  bisected  at  the  given  point. 

2.  The  altitude  is  equal  to  the  diameter  of  the  base. 

3.  The  side  parallel  to  the  wall  is  double  each  of  the  others. 

4.  The  diameter  of  sphere  =  edge  of  cube. 

5.  Three-fourths  slant  height  of  cone. 

6.  Area  is  — .  7.   5  Vi  inches.  8.   3  inches.  9.    —  • 

2  ^  V3 

10.    (at  +  6l)f.  11.    JL.  12.    Arc  =  2  TrrCl  -  V|). 

\/3 

13.  Circular  arc  is  double  the  radius. 

14.  — ;- ,  D  being  the  distance  between  the  centers  of  the  spheres. 

rt  +  i?l 

16.    Angle  at  center  of  variable  circle  defined  by  ^  =  cot  d. 


306  ANSWERS 


Page  111.     Exercises  on  Chapter  VII 


3. 

.00145. 

9. 

2ab. 

6. 

miles  per  hour. 

V2 

10. 
11. 

±2. 

5  IT. 

7. 

The  point  (3,  6). 

12. 

2. 

8. 

At  60°. 

13. 

1  and  5. 

16.  s  =  ^o,   t  =  ^. 

64'  32 

17.  ^. 
16 

19.    ±  16,  T  12  feet 

per  second. 


Page  116.     Art.  68 
3.    1.  4.    (x-\-y)coaxy.  6.    1. 


Page  120.     Art.  71 


3         ax  +  hy  +  g 


hx  +  by+f  2/3 

Pages  127-128.     Art.  76 


^_2y^  =  0.  8.   ^4-«  = 

dy'^  dy  du^ 


I  1  + 
3.    22  = 


¥i —  ^^'7 

^y"  11.    ^-f^+y=0. 


6.  ^  +  M  =  0.  +  (1  _  2-2)2  2^  +  ^4  =  0. 

Pages  131-133.     Art.  79 

1    ?1^4.M=l  4.    (a)  X -I- 2  2/ =  4  a, 

'    a-^       &2        '  2/-2a;4-3a  =  0. 

y_y,  =  «!yi(x_xO.  (^)2y=±(x+l), 

2.   y  =  x.  (7)  y  =  ic +p,  a;  +  y-3p  =  0. 

8.   2  y  =  9  X  -  3,  9  y  +  2  X  =  29.  6    3.  6.   4  Vl?. 

7.  («)  Parallel  at  points  of  intersection  with  ax  +  hy  =  0. 

Perpendicular  at  points  of  intersection  with  hx  -\-  by  =  0. 

(/3)   Parallel  at  (— ^,  ^ap/2\  .  perpendicular  at  x  =  0. 

(7)    Parallel  at  ( i^    2Via\  j   perpendicular  at  (0,  0);  (2  «,  0). 
V  3'       3     1 


ANSWERS  307 


8.    1  —  1  = ,  i.e.  they  must  be  conf ocal. 

a     b     a'     b' 


9.  ^.  11.  ^.  12.  y^. 

2.2  nx 

13.   ^<^  +  «.  19.    (2j9,  ±2joV2). 

o 

Pages  136-137.     Art.  82 

1.  rp  =  d. 

2.  Polar  subtangent  =  ^,  Polar  normal  =  Va"-^  +  p^,  Polar  subnormal  =  a. 

a 

3.  ^  =  -  +  2  ^,    Subtangent  =  -  p  cot  2  ^,    Tangent  =       "^      , 

Subnormal  =  -  «!j1H^,  Normal  =  «^. 
P  P 

6.  -,2  a  sin2  ^  .  tan  -• 
2  2  2 

7.  They  have  a  common  tangent  at  the  pole  ;  elsewhere,  -• 

Page  142.    Exercises  on  Chapter  XI 


a    a 


1.    A^-i-?.  2  Vox,  4  7rVa2  +  ax,  4  Trax.-  2. 

8.    secx.  5.   7r-^(a2_a;2).  7-    ^2  ap. 


4.    7r( 


6.   /)Vl  +  (loga)2. 


Page  151.     Exercises  on  Chapter  XII 

1.  y  =  0,  X  =  a,  X  =  —  a.  8.   x  =  0  twice  ;  one  parabolic 

2.  x  =  0,  x  =  2a,  y  =  a,  y  =  -a.  branch. 

-  ^       .        .  9.    X  =  0,  «  =  0,  X  +  w  =  0. 

3.  y  =  a,  «/ =  —  a ;  two  imagmary.       ,^  '  •'^       ; 

10.    2/  =  X  ;  two  imagmary. 

4.  y  =  a  ;  X  =  c  twice.  ^^    x  +  ?/  +  a  =  0 ;  two  imaginary. 

5.  ?/  =  -  X  +  - ;  two  imaginary.  12.    y  +  x  =  0  ;  two  imaginary. 

13.    X  =  0  twice  ;  x  =  y,  x=:—  y. 

6.  X  =  1 ;  one  parabolic  branch.  14,    y-,^^y-_^.  ^^q  imaginary. 

7.  x  =  —  a,y  =  —  b,y  =  x  +  b  —  a.      15.   x+2  2/=0,  x+2/  =  l,  x— 2/=  — 1. 

Page  158.     Exercises  on  Chapter  XIII 

1.    An  inflexion  s^x  =  y  =  2. 

8.  Point  of  inflexion  at  (a,  f  a) ,  tangent  is  x  +  y  =  -^  •    Bending  changes 
from  negative  to  positive. 


308  ANSWERS 

Pages  162-163.     Art.  102 

1.  First.  5.    y+l2x  =  10. 

2.  They  do  not  touch.  6.   Second. 

3.  Third.  7.    a=-l. 

4.  3  x(x  —  a)  =  a(^y  —  a). 


1.   2  a. 

3.    c». 


4    (^i±i!)|. 


Page  168.     Art.  108 

5     (c2a;2-a2)l 

8. 
9. 

3(axy)i 

ay/xi%a  -  3 a;)* 
3(2  a  -  xY 

7     2(a:  +  y)i 
Va 

10. 

a 

Page  170.     Art.  109 

-  -S- 

e   4V^ 

3     ' 

7.   «(^  + 

<.)! 

1.  pVl  +  (loga)2. 

2.  «i. 

3p 

3  q  (5  -  4  cos  g)t 

,9  -  6  cos  ^ 

4  2p!. 

Page  176.     Art.  HI 

1.  «  =  o,  /3  =  o.  r?     _?\ 

4.    «  =  a;-^-\,e--c''j,  /3  =  2y. 

^a4  4-15y4           a4y-9y^  6.    (au)f  -  (6^)1  =  (a^  +  6^)1 

•"         6a^y     '  ^          2  a*      *  7.   (a  + /S)^  (a  - /3)^  =  2  ai 

Page  186.     Esercises  on  Chapter  XV 

1.  (0,  0) ;  ax±by  =  0.  6.   Two    nodes    at    infinity  ;    the 

«    ^/^  /^^              *«    *!•  J          A  asymptotesare  x  =  y  ±  l,x  +  y  =  ±  1. 

2.  (0,0):  cusp  of  first  k.nd,y  =  0.  ^    („,  _„).  (/„,  q);  (-«,0)  = 

8.   Four    cusps    of     first    kind ;  the  tangents  are,  respectively, 

(0,  ±a),(±a,0);  y  =  0,x  =  0.  V3(y  +  a)  =  ±  v^x  ; 

4.    (0,  0);   conjugate  point  with  2(x  -  a)  =  ±  VSy ' 

real  coincident  tangents,  y  =  0.  8.    (-  a.  0) ;  conjugate  point. 

6.    (0,  a);  y=a4-a; ;  cusp  of  second  9.    (0,  0);  x  =  0,  ?/  =  0. 

kind.  10.    (0,  0)  is  a  tacnode  ;  y  =  0. 


ANSWERS  309 

Pages  193-194.     Exercises  on  Chapter  XVI 

3.  x^  +  y^  =  ci  8.    (c,  c),  (0,  0),  and  one  at  infinity 

4.  4:xy  =  k^-  on  each  axis. 


Page 

201.     Art.  125 

1. 

ixf. 

2. 

a  +  1 

3. 

f.i 

4. 

2  m      m-i 
2m- 1 

6.   51ogx  + 

3 

2x2 

1 

3x8 

6. 

aac  -  1  at  a;^  +  §  < 

airci-^a;2.            7.   \(x^ -^  a^y. 

8. 

(ax  +  6)«+i 

19. 

(a  +  &)»«+»« 

31. 

\  sec8  M. 

a(/i  +  l) 

w  log  (a  +  6) 

32. 

—  log  cosec  M. 

9. 

log(x  +  a). 

20. 

1  (a;  +  sin  x) . 
cos  (m  4-  w)x 

33. 

—  cosw. 

10. 

log  V2  ax  —  x^. 

21. 

34. 

sin-i  -. 

m  +  n 

a 

11. 

log  tan  x. 

22. 

—  ^  cos  X2. 

35. 

isin-i2x. 

12. 

-l0g(l+C0SiC). 

23. 

sin  X  —  ^  sin^  x. 

ltan-i~. 
a           a 

13. 

log  (log  x). 

24. 

— cosx+^cos^x. 

36. 

14. 

flog(x3  +  l). 

25. 

^  X  —  :|  sin  2  X. 

37. 

i-ton-i^. 

26. 

tan  X  —  X. 

a6           a 

15. 

log  sec  X. 

16. 

log  sin  X. 

27. 

\  tan8  X. 
-icot(ax  +  6). 

38. 

itan-%'- 

28. 

17. 

le". 

a 

-f  (cotx)i 

39. 

seo-i  («  -  1). 

a 

29. 

40. 

Isec-'S. 

18. 

\e'\ 

30. 

log  tan  X. 

a         a 

Page  205.  Art.  126 

1.  xsin-ix  +  Vinr^.  6.  secx[logcosx+l]. 

2.  ex  tan-i  e-  -  \  log  (1  +  e2x).  7.  K(«^^  +  1)  cofi  x  +  x\ 

3.  x2sinx4-2xcosx-2sinx.  »•  Ksi"  3x  -  3  xcos3x]. 

9.  icx(sinx  +  cosx). 

4.  _± f  log  x  —  \  • 

w  +  1  V              w  +  1  y  10.  i  e*  (sin  x  -  cos  x). 

6.   ^[2x3tan-ix-x'''+log(l  +  x2)].  11.  fcosxsin  2x  -  ^cos2  xsin  x. 


310  ANSWERS 


Pages  206-209.     Art.  127 

4.   Ksin-ix)2.  14    logtan(^  +  ^V 

6.    icos(x2+l)[l-logcos(x2+l)].  ^2     4/ 

1  ..  16.    J-tan-i2£±i. 

.  x         .     .x-a  18.   -llog?^^. 

10.   vers-i  - ,  or  sm-^ "        »     ^  x  +  1 


a'  a 


11.   log(a;  +  \/x-^±a2). 


19.  |sin-i(3x-5). 

20.  icos(3x-2) 


12.    —log  ^5 — -'  +(a;-f)sin(3x-2). 

2  0!  X  -\-  d  .—- — 

Pages  213-214.     Exercises  on  Chapter  I 


1.    \/x2  +  2  X  +  2  -  log  (X  +  1  +  Vx^  +  2  X  +  2), 


,      1  -  2  X  +  V5  x2  -  4  x+1 
log ^ 

X 


3.    §  V3x2  +  X  -  2  -  -^log  (X  +  ^  +  Vx2  +  ix  -  i). 
^  3V3 


^^:iOg^. 

x-l 


4.  _  V8  4-  4 X  -  4  x2  +  |sin-i 

5.  - V-x2  +  2x  +  1  +  sm-i^^. 

>/2 


2x  — a 


6.    Vr^^  +  8in-ix.  7.    v^^r=:¥2+|sin-i— ^ 

. ,, -,       1    ,      rV2+Vx2+2x4-3~l 

8.  Vx2+2x+3-log(x+l  +  Vx2+2x+3)--— log[ ^  ^_^^ J 

9.  Vx^  +  x+l-^log[x  +  i  +  Vx''  +  x  +  l]-log[^~'^'^^^'^'^^'''^^^ 

1  ^ 14.    -log(c-*  +  v^2*-l). 

10.  log-(x2+l  +  Vx*  +  x«  +  l).  rV2W^T2-| 

16. log • 

11.  c<  2V2        I  «  -• 

n.  iiog^— ^. 

18.   ilog(6x»  +  12x  +  5).  e*  +  l 

18.  ix5tan-ix-,«)yX*  +  TJff«a-Tifflog(x2  +  l). 

19.  «  -  log  (a?  +  1)  +  2  tan-» ». 


ANSWERS  311 

20. -^— -^[2  +  2xloga+(xloga)2]. 

a*  (log  a)  3 

21.   tan^-sec0.  23    1 iog(acos2a;+6sin2a;). 

22     -cot^.  2<^^-«> 

2  24.    ^[x  —  log(sinx  +  cosx)]. 

25.'  i(»^^  +  2)Vx2-l. 

26.  6[^xH  ix  -  ^x^  +  -Jx*  -  1  x^  +  Ixi  -  x^  +  log(xi  +  1)]. 

1    /I i 30.   tan-i(logx).  ' 

27.  -^vl  -logx.  1 

3j    i 

28.  log  Ve^'  +  1.  •    6  (a -ft  tan  x)' 

.0.    Sin-.  (-11^).  »-  i--[^-^^^S^J- 

33.  2  Vx  vers-i-  +  4\/2  a  -  x.  \ 

34.  -lV3x2  +  2x  +  l+logP  +  ^+^-^^^+-^^+l1.  ^ 

X  L  a;  J 

85.  -^.og(.  +  V^^-^r^+J^^. 


Pages  219-222.     E:sercises  on  Chapter  II 


2 


Vx'-^  -  2  X  -  3  -  2  log  (x  -  1  +  Vx2-2x-3). 


3.  ^-^V2ax-x^-f«-sin-i^::i«. 
2  2  a 

5.  The  arrangements  which  can  be  used  are  [5],  [C],  [5],  [O],  and 
[5],  [^],  [C],  [CI 

6.  if— ^— +itan-i^l.  9.   5^'-v/^2Tr^ +  ^sin-i?- , 
^  I  a;2  +  4^          2j  2  2  a. 

7.  _l^.IlJ_+-±_tan-i2^Ill.       10.    -^Z^. 
3(x2-x+l)     3V3  \/3  «'^ 

8.  _^Va-rx-^  +  f  sin-i|.  "      "'    3 , ^.f^  ,)f  +  ^^^Sf^* 

12.  ^(2x2  +  5  a2)  V^M^  +i|liog  (X  +  Vx2  +  «2). 
8  o 

13.  ^  Vx2  +  a  +  -  log  (x  +  Vx2  +  a). 

14.  K2  aj2  -  «a;  -  3  a^)  V2ax-x^  +  ^  sin-i^^=-^. 


15         (2  «x  -  x2)^  Ig         vl  +  a;^ 

3ax8       '  "2x2 


312  ANSWERS 


J-    3(x  +  2)8-5(x  +  2)  I    3  ,      x  +  1 
8(x2  +  4x  +  3)2      ^16     ^x  +  3 

18.    i(x+  l)Vl-2x-x2  +  sin-i^^. 

V2 

Page  226.     Art.  133 

2a        x+a  2  2^x+l 

^    ^^g  x'cx"^- 1  )•  ^'   x  +  \og(x-a)\x-b)\ 

6.   ?-±^  log  (X  -  2  -  v/3)  -  ?—2^  log  (X  -  2  +  V3). 
2>/3  2V3 

^^     ^(2x  +  l)(x  +  2)  °  x-c 

9.  x  +  -^— [a2  1og(x  +  a)-62log(x  +  6)]. 
6  —  a 

10.  log[(x  +  2)V2x-l].  13.   ^_7x  +  641og(x  +  4) 

11.  log(^-«)Cx+ft).  !271og(x  +  3). 

X 

12.  ^log ^ -.  •        14.   JLlog^^. 

*    ^  (2  +  X)  (1  -  x)6  2ab        ax  +  b 

15.  sin-i-^ —  sin-i^^~^)^^  -  log  (x  -  1)"^(2  -  x  +  V2^=^). 

V2      v^  X  -  2 

16.  log  (X  +  2  +  y/x^  +  4  X  +  7)  -  — log(x  +  2)"\  V3  +  Vx2  +  4x  +  7) 

V3 

+  log  (x  +  3)"\l  -  X  +  2Vx2  +  4  X  +  7). 
Page  227.     Art.  134 
2. +^log^^^±-l.  9.   ax-i  +  log 


2(x-l)  X— 1  X  x  +  a 

8.   x-51og(x-3)-p^.  10.   x+-^-K281og(x+3)-logx]. 
^—^  3x 

2(a2-xa)  11-   -a^og  

-1 


X 


V2x(x  +  V2)  l**-   2  log[x  -  1  +  Vx^-2x  +  2] 
«•   f-2x  +  ^^«+.og.(.+l)'.  _.„,[-lW.»_-2x^2-| 

7.  log(x3-a2)--,^^.  +-^  Vx^-2x  +  2. 

x*  —  a*  X  —  1 

8.  —7-^^ 18.  log(x-a)  +  i^5!_=_i«?. 

4(>/2  +  l-x)2  ^^'^       ^      2(«-a)« 

\ 


ANSWERS  313 


Page  228.     Art.  135 


4.   3^,[log(a=  +  a)  7. 

-^log(a;2-ax+a2) 


tan- 


V3  2  a;2  +  1 


,    /o.       i2a^-an  8. l_tan-i-. 

+  V3tan-i— — ^J.  2a{x-a)      2  a^  a 

6.    -Itan-i^  +  ltan-i?.  9.    x-log^'+^^  +  ^ 

a  a     b  b  x—1 

Page  230.     Art.  136 

2.  tan-ia;  +  — ^^ •  6.   _— ^— _  +  log (x^  +  a^) 

x2  +  l  2(aj2  4.a2;        ^^  ^ 

3.  Atan-i^+    ^-^^^  .  -i-tan-i?. 
2  a           a     2  (x2  +  a2)  2  a  a 

4.  ilog  ^^  +  ^    +     ^-1    . 
*       (x+l)2     2(a;2+l) 

^_l±2^_3tan-ia;. 
2(x2  +  l) 

Page  232.     Art.  138 

2.    -2Vx-nx^  +  12  x^  +  6  log (x^  +  l)-12tan-ixi 


3.    logX^LzJ:.  4.    2v^-3v^x4-6v^x-61og(v^+l). 

V  X  +  1  +  1 

5.    21og(Vx^4-l) =Ji 

Vx  -  1  +  1 


6.    2tan-iV^r3^.  7.    llog^^~^~A 

^        Vx  -  a  +  6 

8.    14(xT^?  _  1  x7  +  i  a;T?  -  1  x?  +  ^  xT?). 


Page  236.     Art.  139 

1)        . 


3.  21ogri-At^l  + ^ 

L  ^l-xj      x-l-\- 

4.  _21og[V2+V^ff3j. 


\/l^^x2 


314 


ANSWERS 


1. 

3. 

5. 
6. 

7. 
8. 

11. 
12. 


Pages  236-237.     Exercises  on  Chapter  IV 


2.   |(a;-a)7- 


2(a;2+  l)\/x2  +  2 


^[x2  -  a;Vx2  -  1  +  log  (a;  +  Vx^  -  1)]. 


4V2        V2(x-a)i+l 


^a;(Vx2  +  2-Vx2+  l)+^log 

T3^(2x-3a)(a  +  x)i 
61og(x3- 3x^  +  5). 


X  +  Vx=2  +  2 


9.    


ti 


p 


-4 


V  Vx+  1  +Vx-1 


10. 


log 


Vx2  -  a2  +  a;V2 


2v^a2       Vx2-a2-xV2 
f  x^  - 1  xs  +  f  x^  +  2  x^  -  3  x^  -  6  x^  +  3  log  (xi  +  1)  +  6  tan-i  xi 

Page  239.    Art.  142 

\  tan^  X  +  tan  x.  5.  f  cosec^  x  —  cot  x  —  §  cot*  x. 

-^cot^x-cotx.  6.    -64[cot4x  +  |cot84x]. 

tan  X  +  I  tan^  ic  +  i  tan^  x. 
-  128  [cot  2  X  +  cot8  2  X 
+  |cot62x+  }cot7  2x]. 


-—  +  log  tan  X. 

2  tan2  X 

I  C0t8  X  —  ^  COt^  X. 


Page  240.    Art.  143 

\  sec*  X  --  I  sec2  x.  6.   ^  sec*  x  —  sec^  x  +  log  sec  x. 


I  cosec''  X  -\- 1  cosec^  x 
^  cosec*  X. 


6. 


sec^^x     sec*»  ^^ 


3.  -  {\  sec^  aa;  —  f  sec*  ax  +  sec  ax). 
a 

4.  —  (sin  X  +  cosec  x). 


-  1         «  -  3 

7.  log  sec  X. 

8.  —  log  cosec  a^ 


Page  241.     Art.  144 


1.  —  I  cot'  X  +  cot  X  +  X. 

2.  —  tan*  ax log  sec  ax. 

2a  a 

8.    i(tan2x  +  cot2x) 

+  4  log  (sin  X  cos  x). 


^   tan"  ^x 

71-1   * 

6.   \  tan'  X  -  J  tan»  «  +  ^  tan»  x 
—  tan»  +  «. 


•1'- 

ANSWERS  315 

Pages  242-244.    Art.  145 

2.  -  cos  X  +  ^  cos3  X.  16.   j^s  (5x4-1  sin^  2  x  —  sin  4  x 

3.  -  ^  cos^  X  +  ^  cos^  X.  —  |sin8x). 

4.  log  sin  X  —  sin2  x  +  ^  sin*  x.  17.   ^i^  (3  x  —  sin  4  x  +  i  sin  8  x). 

5.  fcos4x  +  3costx-|cosix.         18-   i(3x  +  sin  4  x  +  |sin  8x). 

^     ^  ,,  3      „  ,^  5         19.    —  icotx  —  icot^x.  u 

6.  4(1 -cos x)^  -  1(1 -cos x) 2.  2  ^ 
'\       ,       ^        ^^                           20.   logtan2x. 

8.    -icot^x.  ^ 


21.  tan  X  +  ^  sin  2  X  —  f  X, 

22.  2cotx-^cot8x+|x+|sin2x. 


9.    —  cot  X  —  I  cot^  X  —  I  cot*^  X 

10.  -  C0t5  X(i  +  }  C0t2  X). 

11.  -4cot3x-2cotx  +  tanx.  23. L(  ^^'^~^)  . 

12.  I  Vtan  X  (tan  x  —  3  cot  x).  ~  , «4  r 

^   -  ,                  ,    ^  24.   ?(2x2-a2)Va2-x2+-sin-i?. 

-„    tan»-^x     tan"+^x  8                                     8          a 


^  ^+^  o^     Vx2-a2        1    ^        a; 

25. sec^— 

15.    I  X  -  s\  sm  4  X.  2  a2  a;'2        2  a^  a 

Pages  245-246.     Art.  146 

1.    J_tan-if«i^Il^V  ^tanfx-^V 


Va2  +  62 


2V3       V3tan^x-^)4-l 


6tan?-a-\/a2  +  62  -    ^    in^^^"^^-^-^ 


log 


log ____.  2  VS       tan  X  -  2  +  V3 

6  tan  -  -  a  +  v'c2  +  62 


3.  itan-i(*-?|^y 

4.  itan-ihtan  (7--)]" 


a(a  tan  X  +  6) 

7.    -L  tan-i  (*^2^V 
V2  V  V2  / 


Page  247.     Exercises  on  Chapter  V 

8.  e^fsin  ?+cos?V 

9.  —  ^  e-*(sin  x  +  cos  x).   1 

5.    itan2x(2  +  tan2x).  !<>•    i  e2x(2  -  sin  2x  -  cos2x). 

1  /^     _.\  11.    ie*(sinx  +  cosx  —  |sin3x 

«•    -iif.  +  '^^'^d+i)-  -icos3.). 

^     ,  .     „      .  -  -     sin«+i  X      sin"+3  x 

7.    i  tan2  X  sin  x  16. —  • 

n+1  n  -{-  S 

4frsin«-logtanf-  +  -l    .  ,»     „       2         «   / 

^L  \2      4yj  16.    f tan5x-2Vcotx. 


316 


ANS WEBS 


17.  -32cot2a;(l +  |cot22x+ ^cot*2x). 

18.  ^tania;(H- f  tan-2^x  + ^tan^^x). 

Page  259.     Art.  152 
6.   2. 


2a 

4ct68 
3 


7.   00, 


Pages  262-263.     Art.  154 
2.    2a6.  3.   2.  4.   20  V5^.  6.   2  a.  7.    12. 


l      9.    |.      10.   faVop;  |p2.      n.   ^a2.      12.   4a2tan-i-^;  4  Tra^. 

^  a 


13.    7ra6. 


4.   X 


14.  f7ra2. 
Page  266.     Art.  155 
6.    3  7ra2.  6.   4a2.  7.    00. 

Pages  268-269.     Art.  156 
Stto^^  4.    ^a(pi-p2). 


15. 


a"^  -  a^i 


log  a 
8.   V-  9-  ^<1' 


5. 


4a2 


3 
6.   7ra2. 

Pages  270-271.    Art.  157 


1.  i)[\^4-log(l+V2)]. 


61a 
216* 

6  a. 

2irr. 


6.   |(e"-e    '»). 


6.   2  -  \/2  +  log 

^    4(a8-&8) 
a6 


l4-\^ 
V3    * 


8  a. 
2ira 


Page  272.    Art.  158 
8. 


.    2arV6-2-V31og     ^  +  ^   1 
L  \/2(2+V3)J 

aftan  -sec-  +  logf  tan  -  +  secf^  1*'. 
L       2       2         *V       2  2yjtf, 

a[--  vTT^  +  log  (d  +  >/rT^)T'. 


Pages  273-274.     Art.  159 


8  a. 


8  ma 


Kxif  +  yit)i-|. 


8.   6  a.  4.   ia^i«. 

7.    >/2(«*>-  1). 


ANSWERS  317 

Pages  276-277.    Art.  160 

..(.-2)/^  4.fC3V2-logCl.V2)3. 


(a)  2  7r6f6+ C0S-1-). 

(6)  2  ,ra2  +  _^^^  log  r«-+  ^«'-^'' 


Va2  _  62. 


4  7ra2. 

9. 

6. 

7.  (a)  7r6Va-^  +  62. 

«i«^. 

8. 

(/3)  7raVa2  +  62. 

3 

Pages  278-279.    Art.  161 

1. 
2. 

4  7r«2  6 

3 

4  7rr3 
3 

3.    TTk^;  00. 

-     32  7ra3 
•      105 

5. 
6. 

7ra3^ 
15* 

4  7r2a8. 

7. 

TrfSa^ 

log 

2a_ 

-4a2(2a-yi)];  oo. 

8. 

5  7r2a3. 

3    8.a« 

10. 

-'-^  fi. 

3  6V2 

Pages  280-283.     Art.  162 

1.    f  TT  ahc.  2.    1 TT  a6.  5.   41  cu.  ft.  7.  —  cu.  ft. 

V3 


3.   \Ah.  4. 


■n  abp 


0 


9,   p=  ea' 


■  c 


12.    ^Tahc^. 

Pages  284-285.     Exercises  on  Chapter  VII 
1.    a21ogw.  2.    1.  5.   a2(2+-j.  6.    j%. 

3.   ^^i|!^l     4.    |.  7.    4a6tan-^        9.    ^h!^ 

Sab  6  a  4  a 


10.  ^ r^  vT+T^  +  log (e  +  vTT^)]^'. 


13.    'La"  [e  a  _  e  a  ]  ^.  ^  a2a;i.  14.  2  7r2a8. 


11.   4  7r2aA;.  12.  2  7r2a2A;. 

4 

15.  2  IT  a2  (3  sin  h-Zh  cos  fi  -  h^  sin  «i). 

16.  ^.  17.   2  7ra2.  18.    alog^. 

3  yi 


318  ANSWERS 

Pages  287-288.     Art.  163 

3.   xy  =  C2/2  +  C'y  +  J.       ^.  y  =  kx(logx-l)  +  CiX-^C2.       b.  ^kA 

Page  292.     Art.  165 
1.  xy  +  O.       2.   -  cos  X  cos 2/+  C.         6.   st^  +  y^  -  S  axy  +  C 

X  X 

3.  Impossible.  4.  log-+C.        6.  tan-i-. 

7.   ^oi^  +  x^y  +  5x  +  ^y^-iy^-\-C. 


^ 


Page  294.     Art.  167 

1.  1.  2.  fa3.  3.   66».  .4. 

Page  295.     Art.  168 

2.  64..  8.    ^-2  Vs. 

o 

Page  298.    Art.  169 
2.    2fl^.  3.   Y^a.  4.   ^.  6.  iirabc.  6.  ^^ 


INDEX 

(The  numbers  refer  to  pages) 


Absolute  value,  59. 
Absolutely  convergent,  59. 
Acceleration,  111. 
Actual  velocity,  105. 
Arc,  length  of,  269. 
Area,  by  double  integra- 
tion, 294. 

derivative  of,  23. 

formula  for,  255,  256. 

in  polar  coordinates,  268. 

in    rectangular   coordi- 
nates, 260. 
Asymptotes,  143. 
Average  curvature,  166. 

Bending,  direction  of,  152. 
Binomial  theorem,  73. 

Cardioid,  area  of,  268. 
Catenary,  168,  283. 
length  of  arc,  271. 
volume   of    revolution, 
285. 
Catenoid,  276. 
Cauchy's  form  of  remain- 
der, 71. 
Center  of  curvature,  163. 
Change  of  variable,  124. 
Circle,  area  by  double  in- 
tegration, 295. 
of  curvature,  163. 
Cissoid,  168. 

area  of,  266. 
Component  velocity,  107. 
Concave,  152. 

toward  axis,  157. 
Conditionally  convergent, 

59. 
Conditions    for    contact, 

161. 
Conjugate  point,  184. 
Conoid,  281. 


Constant,  1. 

factor,  31,  199. 

of  integration,  200. 
Contact,  159. 

of  odd  and  even  order, 
161. 
Continuity,  13,  113. 
Continuous  function,  13. 
Convergence,  57. 
Convex,  157. 

to  the  axis,  157. 
Critical  values,  93. 
Cubical  parabola,  262. 
Cusp,  182. 
Cycloid,  length  of,  273. 

surface   of   revolution, 
277. 

Decreasing  function,  25. 
Definite  integral,  251. 

geometric   meaning  of, 
253. 

multiple  integral,  293. 
Dependent  variable,  1. 
Derivative,  19,  20. 

of  arc,  138. 

of  area,  23, 142. 

of  surface,  140. 

of  volume,  140. 
Determinate  value,  78. 
Development,  56,  80. 
Differentials,  110,  196. 

integration  of,  289. 

total,  117. 
Differentiating    operator, 

24. 
Differentiation,  24. 

of  elementary  forms,  49. 
Direction    of    curvature, 

164. 
Discontinuous  function, 14. 
Divergent  series,  57. 
319 


Ellipse,  area  of,  263. 

length  of  arc  of,  274. 

evolute  of,  178,  284. 
Ellipsoid,  volume,  280. 
Envelope,  187. 
Epicycloid,  length  of,  273. 
Equiangular    spiral,   282, 

284. 
Evaluation,  80,  81. 
Evolute,  170. 

of  ellipse,  176,  271,  284. 

of  parabola,  175. 
Expansion   of    functions, 

56. 
Exterior  rectangles,  254. 

Family  of  curves,  187. 
Formula  for    integration 

by  parts,  203. 
Formulas  of   differentia- 
tion, 49,  50. 
of  integration,  198,  210. 
of  reduction,  217,  218. 
Function,  1. 

Hyperbolic  branches,  143. 

spiral,  area  of,  269. 
Hypocycloid,  area  of,  263. 

length  of  arc  of,  271, 273. 

volume  of  revolution  of, 
278. 

Implicit  function,  120. 

Impossibility     of    reduc- 
tion, 218. 

Increasing  function,  25. 

Increment,  13,  15. 

Independent  variable,  1. 

Indeterminate  form,  77. 

Infinite,  2. 

Infinite  limits  of  integra- 
tion, 257. 
ordinates,  145. 


320 


INDEX 


Infinitesimal,  2. 
Integral,  195. 

definite,  251. 

double,  292. 

multiple,  292. 

of  sum,  199. 

triple,  286,  292. 
Integration,  195. 

by  inspection,  197. 

by  parts,  203. 

by  rationalization,  231. 

by  substitution,  205, 238. 

formulas  of,  198,  210. 

of    rational    fractions, 
223. 

of  total  differential,  289. 

successive,  286. 

summation,  248. 
Interior  rectangles,  251. 
Interval  of  convergence, 

57. 
Involute,  170. 

of  circle,  274,  285. 

Lagrange's    form   of   re- 
mainder, 70. 
Lemniscate,  area  of,  268. 
Length  of  arc,  269. 
of  evolute,  173. 
polar  coordinates,  271. 
rectangular  coordinates, 
269. 
limit,  1. 
change  of,  in  definite  in- 
tegral, 295. 
Limits,  infinite,  for  defi- 
nite integral,  257. 
Logarithm,  derivative  of, 

39. 
Logarithmic  curve,  263. 
spiral,  length  of  arc,  272. 

Maclaurin's  series,  63. 

Maximum,  91. 

Mean  value  theorem,  75, 

267. 
Measure  of  curvature,  166. 


Minimum,  91. 
Multiple  points,  181. 

Natural  logarithms,  40. 
Non-unique  derivative,  25. 
Normal,  129. 
Notation  for  rates,  108. 

Oblique  asymptotes,  147. 
Order  of  contact,  160. 

of  differentiation,  121. 

of  infinitesimal,  8. 

of  magnitude,  7. 
Osculating  circle,  163. 
Osgood,  57. 

Parabola,  171. 

semi-cubical,  262. 
Parabolic  branches,  143. 
Paraboloid,  283. 
Parallel  curves,  175. 
Parameter,  188. 
Partial  derivative,  114. 
Point  of  inflexion,  153. 
Polar  coordinates,  133. 

subnormal,  135. 

subtangent,  135. 
Problem    of    differential 
calculus,  16. 

of  integral  calculus,  195. 

Radius  of  curvature,  164. 

Rates,  105. 

Rational  fractions,   inte- 
gration of,  223. 

Rationalization,  231,  233. 

Rectangles,   exterior  and 
interior,  254. 

Reduction,  cases  of  impos- 
sibility of,  218. 
formulae,  217-218. 

Remainder,  61. 

Rolle's  theorem,  67. 

Singular  point,  179. 

Slope,  21. 

Solid  of  revolution,  140. 


Sphere,  volume  by  triple 

integration,  297. 
Spheroid,  oblate,  276,  278. 

prolate,  276. 
Spiral,  of  Archimedes,  136. 

equiangular,    137,    282, 
284. 

hyperbolic,  269. 

logarithmic,  272. 
Standard  forms,  198,  210. 
Stationary  tangent,  153. 
Steps    in    differentiation, 

24. 
Stirling,  62. 
Subnormal,  130. 
Subtangent,  130. 
Summation,  251. 
Surface  of  revolution,  140. 

area  of,  274. 

Tacnode,  182. 
Tangent,  21,  129. 
Taylor,  62. 
Taylor's  series,  66. 
Tests  for  convergence,  58. 
Total  curvature,  166. 

differential,  117. 
Tractrix,  281. 
length  of,  285. 
surface  of  revolution  of, 

285. 
volume  of  revolution  of, 
285. 
Transcendental  functions, 

38. 
Trigonometric  functions, 
integration  of,  238. 

Variable,  1. 

Volume  of  solid  of  revolu- 
tion, 277. 

Volumes  by  triple  inte- 
gration, 295. 

Witch,  area  of,  263. 
volume    of    revolution 
of,  278,  279. 


1 

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